A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
VIA WEAK KÖNIG’S LEMMA
CHRIS J. CONIDIS
Abstract. We examine and completely classify the reverse mathematical strength of the
following theorems in the context of ω-models (by “ring” we mean “commutative ring with
identity”):
(1) The Jacobson radical of an Artinian ring is nilpotent. (NIL0 )
(2) Every Artinian ring is Noetherian. (ART0 )
(3) Every Artinian ring is isomorphic to a finite direct product of local Artinian rings.
(ARTs0 )
More specifically, we show that (1) and (3) are equivalent to WKL0 over RCA0 and that (2)
is equivalent to WKL0 over RCA0 + IΣ2 . The key ingredient in proving these facts is a new
proof that the Jacobson radical of an Artinian ring is nilpotent via WKL0 . Finally, we prove
a definability result in the more general context of noncommutative rings that distinguishes
the commutative case from the noncommutative case.
1. Introduction
All structures considered in this article will be either finite or countable, and all rings
will be commutative and have an identity element unless we say otherwise. This article is
a sequel to [Con10] in which the author attempted to determine the reverse mathematical
strength of the statement “every Artinian ring is Noetherian.” We assume that the reader
is familiar with [Con10], as well as the basics of computability theory (see [DH10, Nie09,
Soa87, Soa] for more details) and reverse mathematics (see [Sim09] for more details). More
specifically, in [Con10] the author shows that, over RCA0 (recursive comprehension axiom),
the statement “every Artinian ring is Noetherian” implies WKL0 (weak König’s lemma),
while, over RCA0 + BΣ2 , the statement “every Artinian ring has finite length” implies ACA0
(arithmetic comprehension axiom).1 The main theorem of our article proves that, over
RCA0 + IΣ2 , the statement “every Artinian ring is Noetherian” is equivalent to WKL0 . In
other words, we will present a (new) proof that Artinian implies Notherian via the axioms
RCA0 + IΣ2 . This answers [Mon, Question 13]. Recall that IΣ2 is the induction scheme for
Σ02 formulas, while BΣ2 is a bounding scheme for Σ02 formulas (see [Con10, Sim09] for more
details). Also recall that a ring R is Artinian if it satisfies the descending chain condition,
and R is Noetherian if it satisfies the ascending chain condition. For more information on
commutative and noncommutative algebra, as well as the theory of Artinian and Noetherian
rings in these contexts, consult [DF99, Eis95, Lam01, Lan93, AM69, Mat04]. We will assume
that the reader is familiar with the basics of commutative and noncommutative algebra. The
author is currently planning to write a follow-up expository article in which he will explain the
new proofs referred to above, as well as their algebraic and logical significance, to a general
Date: September 12, 2011.
2000 Mathematics Subject Classification. Primary 03F35;
Secondary 03D80, 13E10.
Key words and phrases. Computability theory, reverse mathematics, Artinian ring, Noetherian ring, Jacobson radical.
The author was partially supported by NSERC grant PDF 373817-2009 and a Fields-Ontario Fellowship
from the Fields Institute. He would also like to thank Noam Greenberg for a very productive visit to Victoria
University in Wellington, New Zealand, where most of the key parts of this research was carried out.
1Recently, S. Lempp and K.M. Ng have pointed out that the hypothesis of BΣ is unnecessary. In other
2
words, ACA0 is equivalent to the statement “every Artinian ring is of finite length” over RCA0 .
1
2
CHRIS J. CONIDIS
mathematical audience with a limited background in mathematical logic and computability
theory.
This article is a contribution to the program of reverse mathematics, which began with the
work of H. Friedman [Fri75] and others [FSS83, FSS85] and, generally speaking, attempts
to classify the strengths of mathematical theorems by determining the weakest axioms that
prove them. More specifically, in reverse mathematics one typically attempts to classify the
strengths of “set-existence theorems” from second-order arithmetic by determining the smallest subsystem of second-order arithmetic in which that theorem has a proof. Over the years
five axiom systems have played the most prominent role in reverse mathematics. Indeed, most
theorems from mathematics are equivalent to one of the following five subsystems of second
order arithmetic: RCA0 , WKL0 , ACA0 , ATR0 (arithmetic transfinite recursion axiom), and
Π11 − CA0 (Π11 -comprehension axiom); for more information on the “big five” subsystems of
second-order arithmetic, including their precise definitions, see [Sim09]. Lately, mathematicians have become interested in statements that are not equivalent to one of the “big five”
subsystems of second order arithmetic [Con, DM, HS07, HSS09, Mon06b, Mon08, Nee08] in
the context of ω-models. Recall that an ω-model is a model of RCA0 that has first-order part
equal to the standard natural numbers ω = {0, 1, 2, . . .}. Such models are usually identified
with their second-order parts, and it is well-known that M is an ω-model of RCA0 if and only
if M ⊆ P(ω) is a Turing ideal - i.e. if M ⊆ P(ω) is closed under Turing reducibility ≤T and
join ⊕ (see [Sim09, Soa87, Soa] for more details). Here P denotes the power set operator.
However, it is still unknown whether or not there exists a natural algebraic statement2 that
is not equivalent to any of the “big five” subsystems. Throughout this article we will assume
that all subsystems of second order arithmetic imply RCA0 .
Recently, Montalbán [Mon] has called a theorem of mathematics nonrobust whenever
small “perturbations” of that theorem are equivalent to the original theorem over RCA0 (in
the context of reverse mathematics). He also points out that, usually, nonrobust theorems
correspond to statements not equivalent to any of the “big five” in the context of reverse
mathematics. In [Con10] the author constructs many natural statements about Artinian
rings, each of which is equiavlent to WKL0 over RCA0 and/or RCA0 + IΣ2 (see [Con10,
Theorem 3.4] and its proof for more details), and also shows that the statement “every
Artinian ring is of finite length” is equivalent to ACA0 over RCA0 (+BΣ2 ). This is evidence
that the statement “every Artinian ring is Noetherian” is nonrobust, and thus may not be
equivalent to any of the “big five” subsystems, even in the context of ω-models. If this were
the case then “Artinian implies Noetherian” would have been the first algebraic statement
with this property. Unfortunately, as we said earlier, this is not the case because we will
present a new and unexpected proof that every Artinian ring is Noetherian in the axiom
system WKL0 + IΣ2 .
Recall that the Jacobson radical of a (possibly noncommutative) ring R is the intersection
of all maximal ideals of R, and if R is commutative then for any given x ∈ R the annihilator
of x is the ideal in R defined by
Ann(x) = {y ∈ R : x · y = 0}.3
We will actually present two new proofs that Artinian rings are Noetherian over RCA0 + IΣ2
(in the context of reverse mathematics). Both proofs consider annihilators of various elements
of the Jacobson radical and use the key fact that if x0 , x1 , . . . , xk ∈ R, k ∈ N, are finitely
many elements of a commutative ring R, then the annihilator
Ann(x0 , x1 , . . . , xk ) =
k
\
Ann(xi ) ⊂ R
i=0
2By “natural algebraic statement” we mean a known mathematical statement/theorem about groups,
rings, fields, or their actions.
3In the noncommutative setting one defines left and right annihilators analogously.
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
3
is ∆01 -definable (i.e. computable), uniformly in the input x0 , x1 , . . . , xk , k. However, we consider our first proof to be the main (i.e. better) proof, and it uses the surprising key lemma
that WKL0 proves that the Jacobson radical of an Artinian ring is nilpotent. The most interesting part about our proof is that the new ingredients are purely algebraic, and not logical
or computability-theoretic. In other words, we will present a new algebraic approach to proving the nilpotence of the Jacobson radical in commutative Artinian rings. This approach is
different than anything presented in the standard algebraic literature that we are familiar
with, such as [DF99, Eis95, Lam01, Lan93, AM69, Mat04]. We will also use our proof of
the nilpotence of the Jacobson radical in (commutative) Artinian rings via WKL0 to go on
to prove the full structure theorem for commutative Artinian rings (i.e. every commutative
Artinian ring is isomorphic to a finite direct product of local Artinian rings) in WKL0 . Again,
this is different than anything we have seen in the standard algebraic literature. In particular, our proofs of the “Noetherianess”, structure, and nilpotence of the Jacobson radical of
commutative Artinian rings each avoid constructing any sort of minimal ideal. On the other
hand, all other (standard) proofs of these facts (that we know of) involve the construction
of a minimal ideal of some kind, which (on the surface) requires the stronger axiom system
ACA0 . We see the use of annihilators and the lack of minimal ideals as the key differences
between our (weaker) proof and the other (more powerful) standard proofs in textbooks etc.
A particularly interesting consequence of our new proof is that there is a model of RCA0 in
which every commutative Artinian ring is Noetherian, but not every commutative Artinian
ring is of finite length (as a module over itself). Namely, every ω-model of WKL0 + ¬ACA0
has this property.4 This is interesting in light of the fact that, in the context of commutative
rings, every proof that Artinian implies Noetherian given in the standard texts such as
[DF99, Eis95, Lan93, AM69, Mat04] actually proves that every Artinian ring is of finite
length.
We now introduce several subsystems of second order arithmetic, some of which we have
previously discussed and will play major roles throughout the rest of this article. We take
each of the following statements to imply RCA0 , as well as the statements written beside
them. For more information on subsystems of second order arithmetic, consult [Sim09].
ART0 :
NIL0 :
ARTs0 :
ARTn0 :
NILn0 :
Every Artinian ring is Noetherian.
The Jacobson radical of an Artinian ring is nilpotent.
Every Artinian ring is isomorphic to a finite direct product of local Artinian rings.
Every (possibly) noncommutative left Artinian ring is left Noetherian.
The Jacobson radical in a (possibly) noncommutative left Artinian ring is nilpotent.
The main theorems of this article appear in Section 3 and say that ART0 is provable in
WKL0 + IΣ2 , while NIL0 and ARTs0 are provable via WKL0 . In Section 4 we use the previous
results of the author in [Con10], as well as some new theorems, to show that NIL0 and ARTs0
imply WKL0 over RCA0 . By [Con10, Theorem 4.1] it will then follow that ART0 is equivalent
to WKL0 over RCA0 + IΣ2 , while NIL0 and ARTs0 are each equivalent to WKL0 over RCA0 .
Another interesting aspect of our main theorems is that NIL0 , ART0 , and ARTs0 are some
of the first nontrivial examples of Π12 -statements5 from second-order arithmetic of the form
(∗)
∀(∀ → ∀) or ∀(∀ → ∃∀),
where the ∀, ∃ quantifiers range over second-order (i.e. set) variables and we have omitted
the first-order quantifiers and most of the other aspects of the formulas, to have their reverse
4Or,
more specifically, every low ω-model of WKL0 has this property.
speaking, a Π12 statement is a statement of the form ∀X∃Y · · · , where the variables X, Y ⊆ N
represent sets and the · · · only mention X, Y and other number variables a, b, c, . . . ∈ N. For more information
on second-order arithmetic and Π12 -statements, see [Sim09]. Mathematical statements studied in the context
of reverse mathematics are usually Π12 -statements.
5Generally
4
CHRIS J. CONIDIS
mathematical strengths completely determined in terms of every other subsystem of secondorder arithmetic whose strength is currently determined.6 Other statements of this form
(∗) have been examined by Montalbán [Mon06a] and by Downey, Hirschfeldt, Lempp, and
Solomon [DHLS03] (see these references for more details) in the context of extendibility of
orderings ω (the natural numbers) and η (the rational numbers), and in each of these cases
the precise reverse mathematical strengths of the theorems considered is still open. One of
the reasons why statements of the form (∗) are difficult to classify in the context of reverse
mathematics is because, generally speaking, it is difficult to deduce consequences from (i.e.
code information into) such statements. This is because the implication → is actually a
disjunction, and to obtain a useful logical conclusion from a disjunction A ∨ B, one usually
proves the negation of one of the disjuncts, say ¬A, and then concludes B. In our case, since
we are working over RCA0 , one must prove ¬A in RCA0 to conclude B, and, since RCA0 is
a very weak axiom system, this can be quite difficult or even impossible. More generally,
however, if a mathematical logician is in the nontrivial case when the standard proof of a
given theorem is not obviously valid in RCA0 , or, more generally, the theorem does not easily
reverse to the standard proof (as in our case), then he or she has the daunting task of either:
(1) Deducing a nontrivial consequence from the theorem that reverses to a standard
proof; or
(2) Finding a new proof of the theorem that is logically different from any of the standard
proofs that appear in the literature.
Sometimes, as in our case, the logician must accomplish both of these tasks. In our case task
(1) was essentially achieved in [Con10, Theorem 4.1], while task (2) is the main goal of this
article. We are finally ready to proceed with this task and achieve our main goal.
2. Background, Definitions, and Notation
In this section we introduce our basic notation and definitions, as well as the author’s and
others’ previous results that we will require from [Con10, DLM07] and commutative algebra.
2.1. The basics.
2.1.1. Computability theory. We briefly review the basic definitions and notation that we
require and will use throughout the rest of this article. For more information on the basics
of computability, including our definitions and notation described below, consult [DH10,
Sim09, Soa87]. Most of what follows in this subsection can also be found in [Con10]. Recall
that ω = {0, 1, 2, . . .} denotes the standard natural numbers. On the other hand, N will
denote the first-order part of a (possibly nonstandard) model of RCA0 . We will say that
a property holds for almost all n ∈ N whenever that property holds for all but finitely
many n ∈ N. Throughout this subsection all of our definitions will be made in RCA0 . In
other words, throughout this subsection we will assume that we are working in a possibly
nonstandard model of RCA0 . Lower case roman letters a, b, c, . . . will usually denote firstorder (i.e. number) variables, while capital roman letters A, B, C, . . . will usually denote
second-order (i.e. set) variables. Let A, B ⊆ N. We write A < B to mean that every number
in A is strictly less than every number in B. We say that a function f : N → N is total
whenever the domain of f is N, and we say that f is partial to indicate that the domain of
f may not be all of N. Also, we say that A is computable whenever there is an algorithm
that decides, for each x ∈ N, whether x ∈ A. It is well-known that A is computable iff A is
∆01 -definable, and we will sometimes use the term computable to mean ∆01 -definable. More
information on definability and the complexity of formulas can be found in [Sim09]. One
can also define what it means for a set A to be computable relative to an oracle B [Soa87,
6Richard
observation.
Shore was the first person to point this out to the author who thanks Shore for this astute
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
5
Chapter III]; in this case one usually writes A ≤T B, and ≤T is a quasi-ordering on P(N)
while the relation
A ≡T B, i.e. A ≤T B & B ≤T A,
is an equivalence relation on P(N). The equivalence classes of the ≡T relation are called
Turing degrees. It follows that A is computable relative to B iff A is ∆01 -definable in the
parameter B. We say that a total function f : N → N is computable (relative to B) whenever
there is an algorithm (with oracle B) that, on every input x ∈ N, outputs f (x). It follows
that a set A is computable (relative to B) iff its characteristic function is computable (relative
to B). Via a computable 1-1 and onto pairing function h·, ·i : N × N → N we may speak of
computable subsets of
Nn = N
| ×N×
{z· · · × N}, n ∈ N,
n
and it follows that f : N → N is computable (relative to B) iff the graph of f is computable
(relative to B). We will write A ⊕ B to denote the disjoint union of A and B, i.e.
A ⊕ B = A × {0} ∪ B × {1}.
Similarly, we have that
A0 ⊕ A1 ⊕ · · · ⊕ Ak =
k
[
(Ai × {i}) ,
i=0
and
∞
M
i=0
Ai =
∞
[
(Ai × {i}).
i=0
Recall that an infinite set A is computably enumerable (c.e.) iff
(1) There is an algorithm that lists the elements of A (not necessarily in increasing order);
(2) There is a 1-1 computable function f : N → N such that A is the range of f ;
(3) A is Σ01 definable.
One may also speak of c.e. relative to (the oracle/parameter) B. Recall that there is a
uniformly computable listing of the partial computable functions, {ϕe }e∈N , and that the
Halting Set ∅0 is defined as follows:
∅0 = {e ∈ N : ϕe (e) ↓ },
where ϕe (e) ↓ means that the eth partial computable function halts on input e ∈ N. For
any given oracle B, one can construct an effective listing of the partial computable functions
0
relative to B, {ϕB
e }e∈N , and define the Halting Set relative to B, B , in an analogous fashion.
Let 00 denote the Turing degree of ∅0 . It is well-known that ∅ does not compute ∅0 (i.e. ∅0 is
not computable relative to ∅), and, more generally, for all oracles B, B does not compute
B 0 . We say that a set A is low whenever A0 ≡T 00 . By our previous remarks it follows that
if A is low then A does not compute ∅0 .
denote the set of finite strings formed from elements
Fix a number n0 ∈ N, and let n<N
0
in (the finite set) {0, 1, . . . , n0 − 1} ⊂ N. We will typically use lower case Greek letters to
<N
denote the elements of n<N
0 . For all σ ∈ n0 , let |σ| ∈ N denote the length of σ (i.e. the
number of character bits of σ) and let σ(k), 0 ≤ k < |σ|, denote the k th character bit of σ.
Furthermore, for any given l ∈ N, let
<N
n=l
0 = {σ ∈ n0 : |σ| = l}
and
n0≥l = {σ ∈ n<N
0 : |σ| ≥ l}.
<N
For all σ, τ ∈ n0 , we write σ ⊂ τ to mean that τ is a proper extension of σ (i.e. σ is a proper
initial segment of τ ) and we write σ ⊆ τ to mean that either σ = τ or σ ⊂ τ . Also, for all
σ, τ ∈ n<N
we write στ ∈ n<N
to denote the concatenation of τ to the right of σ, and for all
0
0
6
CHRIS J. CONIDIS
k ∈ {0, 1, . . . , n0 − 1} we write σk ∈ n<N
to be the unique string of length |σ| + 1 that has σ
0
as an initial segment and rightmost character bit k. We say that T ⊆ n<N
is a tree whenever
0
T is closed under ⊂ - i.e. for all τ ∈ T and σ ⊂ τ we have that σ ∈ T . Let nN0 denote the
set of infinite strings formed from elements in (the finite set) {0, 1, . . . , n0 − 1} ⊂ N. We
will typically use the lower case roman letters f , g, and h, to denote elements of nN0 . For all
denote the first l bits of f (i.e. f l is the unique initial
f ∈ nN0 and l ∈ N, let f l ∈ n<N
0
segment of f of length l). Also, for all f ∈ nN0 and σ ∈ n<N
0 , we write σ ⊂ f to mean that σ
is an initial segment of f . Now, if T ⊆ n<N
is
a
tree,
then
we let
0
[T ] = {f ∈ nN0 : (∀n ∈ N)[f n ∈ T ]}
and we say that [T ] ⊆ nN0 is the set of infinite paths through T . Recall that WKL0 is equivalent
to saying “for all n0 ∈ N, every infinite tree in n<N
has an infinite path,” which, over RCA0 ,
0
is equivalent to saying that “for any set A ⊆ N there exists a set B ⊆ A that is of PA
Turing degree relative to A.” Recall that a set B is of PA Turing degree relative to a set
<N
A if every infinite A-computable tree TA ⊆ n<N
0 , n0 ∈ N, (where the finite strings in n0
are coded as natural numbers via some Gödel numbering) has a B-computable infinite path
fB ∈ [TA ] ⊆ nN0 . For nonobvious reasons, this is equivalent to saying that “for every disjoint
pair of computably enumerable sets C0A , C1A ⊆ N relative to A, there is a B-computable
set DB ⊆ N such that C0A ⊆ DB and DB ∩ C1A = ∅.” We will primarily use this (latter)
characterization of “PA Turing degree relative to A” in all that follows. For more information
on PA Turing degrees and their connection to WKL0 , see [Sim09]. Finally, recall that ACA0
is equilvalent to the statement “for every set X, the Halting Set relative to X, X 0 , exists”
and that, for all n ∈ N, the Σn induction scheme, denoted IΣn , is equivalent to the Πn
induction scheme (i.e. induction for all Πn formulas) and to the well-ordering principle for
Σn /Πn formulas. See [Sim09] for more details.
2.2. Commutative and noncommutative algebra. We now explain the basics of commutative and noncommutative ring theory that we will use throughout the rest of this article.
All of our rings R will have an identity element 1 ∈ R. Recall that a computable (possibly
noncommutative) ring is a computable subset of N, endowed with the structure of (possibly
noncommutative) ring such that the addition +R and multiplication ·R operations on R are
computable functions. If R is noncommutative then we use the term “ideal” to mean “twosided ideal.” By local ring, we mean a ring R with identity 1 ∈ R and a unique maximal
ideal M ⊂ R. Let R be a commutative ring with identity 1 ∈ R. Then, if S ⊆ R, we write
hSiR ⊆ R to denote the ideal generated by S in R. Recall that hSiR is the set of finite sums
of the form
k
X
ri xi ,
i=0
where k ∈ N, ri ∈ R, and xi ∈ S. For any k ∈ N and x0 , x1 , . . . , xk ∈ R, we write
hx0 , x1 , . . . , xk iR to denote the ideal generated by {x0 , x1 , . . . , xk } ⊆ R. We sometimes omit
the subscript R in hSiR if it is clear which ring R the set S belongs to. For all A, B ⊆ R, let
SAB = {x ∈ R : x = ab, a ∈ A, b ∈ B},
SA,B = {x ∈ R : x = a + b, a ∈ A, b ∈ B},
and define
A · B = AB = hSAB iR and A + B = hSA,B iR .
Note that AB ⊆ A, B, while A, B ⊆ A + B. We will write A + B = C to mean that
(A+B) = C; in other words, every element of C can be expressed as an R-linear combination
of elements in A ∪ B. Also, for all sets S ⊆ R and ideals I ⊆ R let
(S : I) = {r ∈ R : (∀s ∈ S)[r · s ∈ I]}.
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
7
It follows that (S : I) ⊆ R is always an ideal (since I is an ideal). We will write (x : I) to
mean ({x} : I), x ∈ R, I ⊆ R an ideal. For all multiplicatively closed subsets S ⊆ R not
containing 0 ∈ R, let R[S −1 ] be the ring of fractions given by
nr
o
R[S −1 ] =
: r ∈ R, s ∈ S .
s
Recall that x ∈ R is nilpotent if there exists n ∈ N such that xn = 0. Similarly, we say that
A ⊆ R is nilpotent whenever there exists n ∈ N such that
An = A
| ·A·A
{z· · · · · A} = 0.
n
This is equivalent to the existence of n ∈ N such that for any a0 , a1 , . . . , an ∈ A we have that
n
Y
ai = 0.
i=0
Recall that for all x ∈ R, the annihilator of x, denoted by Ann(x) ⊂ R, is the ideal of R
given by Ann(x) = {y ∈ R : xy = 0}. Similarly, for all x0 , x1 , . . . , xk ∈ R, k ∈ N, we have
that Ann(x0 , x1 , . . . , xk ) = {y ∈ R : (∀i ≤ k)[xi y = 0]}. If R is a quotient ring of the form
R0 /I, for some commutative ring R0 and ideal I ⊆ R0 , then we will write x ∈ R, x ∈ R0 ,
to denote that x ∈ R is the image of x ∈ R0 under the canonical map R0 → R. If I0 ⊇ I is
an ideal of R0 , then we will also write I 0 to represent the unique ideal corresponding to I0
under the canonical map R0 → R. Given two ideals I, J ⊆ R, let
(I : J) = {r ∈ R : (∀x ∈ I)[rx ∈ J]}
and similarly, for all x0 ∈ R and J ⊆ R an ideal, let
(x0 , J) = {r ∈ R : rx0 ∈ J}.
Let
Z∞ = Z[X0 , X1 , X2 , . . .]
be a computable presentation of the standard free commutative polynomial ring over Z with
infinitely many indeterminates X0 , X1 , X2 , . . . and no relations between them. Let F denote
the field of fractions of Z∞ , and let Zk = Z[X0 , X1 , . . . , Xk ], k ∈ N. Furthermore, if R is a
ring and X is a set of indeterminates, none of which appears in R, then let R[X] be the ring
of polynomials with indeterminates in X and coefficients in R. Let
Q∞ = Q[X0 , X1 , X2 , . . .]
be a computable presentation of the standard free noncommutative polynomial ring over Q
with infinite many indeterminates X0 , X1 , X2 , . . . and no relations between them. Now, let R
be a quotient ring of Q∞ , i.e. we have that R = Q∞ /I, for some ideal I ⊆ Q∞ . We say that
x ∈ R is a monomial whenever x is the image of a monomial m ∈ Q∞ under the canonical
map Q∞ → R - i.e. we have that m = x. Furthermore, we say that x ∈ R is a left monomial
multiple of y ∈ R whenever there is a monomial m ∈ Q∞ such that m ·R y = x. Similarly,
we can define right monomial multiple. We also say that x ∈ R is a left divisor of y ∈ R
whenever there exists a ∈ R such that
a ·R x = y,
and we say that x ∈ R is a right divisor of y ∈ R if there exists a ∈ R such that
x ·R a = y.
Note that if x, y ∈ R in the previous sentence are both monomials, then a ∈ R can be taken
to be a monomial as well. We say that x ∈ R divides y ∈ R if there exist a, b ∈ R such that
axb = y.
If x and y are monomials, then a and b can be taken to be monomials.
8
CHRIS J. CONIDIS
2.3. Previous results. In this section we collect various known results that will be useful
later on. The first lemma (Lemma 2.1) is a standard fact from commutative algebra (regarding fraction rings); the second lemma (Lemma 2.2) and its corollary (Corollary 2.3) are
results in computable algebra and reverse mathematics that have essentially appeared in the
literature [Con10, DLM07, FSS83, Sim09] several times but never been explicitly stated as
we will now state them below; the next two theorems are [Con10, Theorem 3.4, Theorem 4.1];
and the last lemma (Lemma 2.7) gives a standard technique for constructing computable
rings (the“pullback technique”).
Let R be a commutative ring (with identity), and U ⊆ R a multiplicatively closed subset
not containing zero. We begin this section by proving an easy and well-known lemma that
we will use in our second proof of Theorem 3.2 in the next section. We use the notation
R[U −1 ] to denote the ring of fractions obtained from R by adding elements of the form u1 ,
for all u ∈ U .
Lemma 2.1. Let ϕ : R → R[U −1 ] be the natural map given by r → 1r , and let I0 ⊂ I1 ⊆ R
be ideals of R such that
x·u∈
/ I0 ,
for all x ∈ I1 and u ∈ U . Then ϕ(I0 )R[U −1 ] ⊂ ϕ(I1 )R[U −1 ], as ideals in R[U −1 ].
Proof. Let x1 ∈ I1 \ I0 , and suppose (for a contradiction) that
x1
x
= ∈ I0 R[U −1 ] ⊆ R[U −1 ],
1
u
for some x ∈ I0 and u ∈ U . It follows that we have
u · x1 = x ∈ I0 ⊂ R,
a contradiction (by hypothesis).
The following lemma and its corollary are from computable algebra has essentially appeared in the literature several times, but never been explicitly stated as follows. Lemma
2.2 and Corollary 2.3 are both well-known by computable algebraists, and we will use them
without necessarily saying so; we have included them mainly for the nonexpert’s convenience.
Lemma 2.2. Suppose that R is a computable commutative ring and {xk }k∈N , S ⊂ R, are such
that, for all n ∈ N, xn+1 is not an R-linear combination of elements in S ∪ {x0 , x1 , . . . , xn }.
Then every PA Turing degree computes an infinite sequence of ideals {Ik }k∈N of R such that,
for all n ∈ N, S ∪ {x0 , x1 , . . . , xn } ⊆ In , but xn+1 ∈
/ In .
We will not prove Lemma 2.2, but the proof can essentially be found in [Con10, DLM07,
FSS83, Sim09]; it was first essentially proven in [FSS83]. The main idea of the proof of
Lemma 2.2 is to construct an infinite computable tree T such that every infinite path through
T computes an infinite sequence of ideals {Ik }k∈N as in the statement of the lemma. The
conclusion of the lemma then follows by one of the characterizations of PA Turing degrees
that we gave above. The following corollary is the reverse mathematical analog of the
previous lemma. It assumes WKL0 .
Corollary 2.3. (WKL0 ) Suppose that R is a commutative ring and {xk }k∈N , S ⊆ R, are such
that, for all n ∈ N, xn+1 is not an R-linear combination of elements in S ∪ {x0 , x1 , . . . , xn }.
Then there is an infinite sequence of ideals {Ik }k∈N of R such that, for all n ∈ N, S ∪
{x0 , x1 , . . . , xn } ⊆ In , but xn+1 ∈
/ In .
Before we proceed we need to recall the following definition from algebra.
Definition 2.4. Let R be a (possibly noncommutative) ring with identity. We say that a
subset S ⊂ R is t-nilpotent if for every infinite sequence of elements x0 , x1 , x2 , . . . , xn , . . . ∈ S
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
9
(with possible repetitions) there exists N ∈ N such that
N
Y
xi = 0.
i=0
Also recall that the nilradical of a commutative ring R is the intersection of all prime
ideals of R, while the Jacobson radical is the intersection of all maximal ideals of R. If R is
Artinian it follows that these two radicals are equal.
For the reader’s convenience we now state the two main theorems that we will use from
[Con10].
Theorem 2.5. [Con10, Theorem 3.4] Let R be a commutative ring with identity. The
following statements are equivalent over RCA0 + IΣ2 .
(1) WKL0
(2) If R is an Artinian integral domain, then R is a field.
(3) If R is Artinian, then every prime ideal of R is maximal.
(4) If R is Artinian, then the Jacobson radical J and the nilradical N of R exist and
J = N.
(5) If R is Artinian, then the Jacobson radical of R is t-nilpotent.
(6) If R is Artinian and the nilradical of R exists, then R/N is Noetherian.
In fact, it follows from the proof of [Con10, Theorem 3.4] that (1)-(5) above are equivalent
over RCA0 . We will refer to and use [Con10, Theorem 3.4] in Section 3 below.
Theorem 2.6. [Con10, Theorem 4.1] There exists a computable (commutative) integral domain R containing an infinite uniformly computable strictly ascending chain of ideals
I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ In ⊂ · · · ⊂ R, n ∈ N,
and such that every infinite strictly descending chain of ideals in R,
R ⊇ J0 ⊃ J1 ⊃ J2 ⊃ · · · ⊃ Jn ⊃ · · · , n ∈ N,
contains a member (i.e. an ideal) of PA Turing degree.
Moreover, in the proof of [Con10, Theorem 4.1] the author proves that there is an infinite
uniformly computable strictly ascending chain of ideals
I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ In ⊂ · · · ⊂ R, n ∈ N,
in R such that
I∞ =
[
In
n∈N
is also computable and every ideal I ⊂ R that is not of PA Turing degree is computable and
equal to In , for some n ∈ N ∪ {∞}. We will use [Con10, Theorem 4.1] in Section 4 below.
Finally, we recall the following useful result in computable algebra that allows one to construct computable rings that are computably isomorphic to computably enumerable subrings
of a larger computable ring. We will use the following lemma in Section 5 below, and its
proof can be found in [DLM07, Section 2.3].
Lemma 2.7. Suppose that Q is a computable ring, and R0 ⊆ Q is a computably enumerable
subring of Q. Then R0 is computably isomorphic to a computable ring R.
2.4. The plan of the paper. In the next section we prove the main theorems of this article
which say that ART0 is provable via WKL0 +IΣ2 , while NIL0 and ARTs0 are provable via WKL0 .
In Section 4 we establish nontrivial lower bounds on the reverse mathematical strengths of
NIL0 and ARTs0 by showing that each of these statements implies WKL0 over RCA0 . It follows
by our previous results in [Con10] that ART0 is equivalent to WKL0 over RCA0 + IΣ2 , while
NIL0 and ARTs0 are equivalent to WKL0 over RCA0 . We now discuss the theorem that we
prove in the last section (i.e. Section 5).
10
CHRIS J. CONIDIS
2.4.1. Section 5: Generalizing our results to noncommutative rings and two-sided ideals. In
the last section of this article we examine a generalization of some of our previous results to
the setting of noncommutative rings and the context of computable algebra and ω-models
of second-order arithmetic. First of all, note that by our results in the next section it follows
that if R is a computable commutative ring in an ω-model of RCA0 with an infinite uniformly
computable strictly ascending chain of ideals
I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ In ⊂ · · · ⊂ R, n ∈ N,
then it follows that R also contains an infinite strictly descending chain of ideals
R ⊇ J0 ⊃ J1 ⊃ J2 ⊃ · · · Jn ⊃ · · · , n ∈ N,
such that the uniform listing of ideals
J=
M
Jn
n∈N
is computable relative to some PA Turing degree. In particular, it follows that one can always
find a sequence of ideals {Jn }n∈N that is uniformly low - i.e. J 0 = 00 . In Section 5 below
we prove a theorem that says that the result of the previous sentence does not generalize to
two-sided ideals in noncommutative rings. More specifically, we prove the following theorem.
Theorem 5.2. Let M be an ω-model of RCA0 . Then M contains a local noncommutative
ring R with unique (computable) maximal ideal M ⊂ R such that:
(1) R contains a uniformly computable strictly ascending chain of (two-sided) ideals
{0} = I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ M ⊂ R;
(2) M is not nilpotent, i.e. for all n ∈ N, M n 6= 0;
(3) Every element of R \ M is a unit (i.e. has both left and right inverses);
(4) R contains an infinite strictly descending chain of (two-sided) ideals (not necessarily
computable);
(5) Every infinite strictly descending chain of ideals in R, J0 = R ⊃ J1 ⊃ J2 ⊃ · · · ,
computes the Halting Set ∅0 (i.e. we have that J = ⊕k∈N Jk computes ∅0 ).
Note that Theorem 5.3 above is purely a definability result and does not have a reverse
mathematical counterpart, because the statements “the Jacobson radical of a two-sided
Artinian ring is nilpotent” and “every two-sided Artinian ring is two-sided Noetherian” are
not theorems of noncommutative algebra.7 Nonetheless, Theorem 5.3 is interesting because
it shows that our results in the next section (when viewed as definability results in the
context of computable algebra and ω-models of second-order arithmetic) do not extend to
the more general context of noncommutative rings and two-sided ideals.
3. Weak König’s lemma and Artinian rings
The main purpose of this section is to show that the structure theorem for Artinian rings
(i.e. ARTs0 ) is provable in the system WKL0 . Along the way we will also show that WKL0 +IΣ2
proves that every Artinian ring is Noetherian (i.e. ART0 ) and that WKL0 proves that the
Jacobson radical of an Artinian ring is nilpotent (i.e. NIL0 ). Afterwards we will give an
alternate direct proof of ART0 via WKL0 + IΣ2 .8 By previous results of the author [Con10]
it will follow that ART0 is equivalent to WKL0 over RCA0 + IΣ2 . In the next section we will
examine the consequences of NIL0 , ART0 , and ARTs0 , in the context of reverse mathematics.
We begin by showing that WKL0 proves NIL0 over RCA0 .
7Although
the corresponding statements “the Jacobson radical of a left Artinian ring is nilpotent” (NILn0 )
and “every left Artinian ring is left Noetherian” (ARTn0 ) are theorems of noncommutative algebra.
8Recall that classifying the reverse mathematical strength of ART was the main motivation for all of the
0
research in this article.
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
11
Theorem 3.1. WKL0 implies NIL0 over RCA0 .
Proof. We reason in WKL0 . Let R be a given Artinian ring. In the proof of [Con10, Theorem
3.4] the author showed that WKL0 proves that the Jacobson radical of R, J ⊂ R, exists. Let
J = {z0 , z1 , z2 , . . .} be an enumeration of the elements of J, and for each n ∈ N, let
An = Ann(z0 , . . . , zn ).
RCA0 proves that the sequence of ideals {An }n∈N exists. Now, suppose that there are infinitely many k ∈ N such that Ank ⊃ Ank +1 . Then, via RCA0 , it follows that there exists an
infinite strictly descending chain of ideals of the form {Ank }k∈N in R, contradicting the fact
that R is Artinian. Hence, there must exist a number n0 ∈ N such that for all n ≥ n0 we
have that
An0 −1 = Ann(z0 , z1 , . . . , zn0 −1 ) = Ann(z0 , z1 , . . . , zn ) = An .
<N
Let T ⊆ n0 be the n0 −branching computable tree defined by
Y
T = {σ ∈ n<N
:
zσ(j) 6= 0}.
0
j<|σ|
First, suppose that T is infinite. Then by WKL0 T has an infinite path f ∈ [T ] ⊂ nN0 such
that for all n ∈ N,
Y
zf (j) 6= 0.
j<n
In other words, J is not t-nilpotent (see [Con10] for more details), which contradicts [Con10,
Theorem 3.4]. Hence, we must have that T is a finite tree.
Now, suppose that T is finite, and let m0 ∈ N be large enough so that there is no string of
length m0 on T . In this case we claim that J m0 = 0. To see
Qwhy, assume (for a contradiction)
that there exist elements x0 , x1 , . . . , xm0 −1 ∈ J such that j<m0 xj 6= 0 and via Σ1 -induction
0
let i0 < m0 , i0 ∈ N, be maximal such that there exists σ ∈ n=i
⊂ n<N
such that
0
0
Y
Y
zσ(j) ·
xj 6= 0.
i0 ≤j<m0
j<i0
0
We claim that i0 = m0 . Suppose for a contradiction that i0 < m0 . Then there exists σ ∈ n=i
0
such that the product
zσ(0) zσ(1) · · · zσ(i0 −1) xi0 xi0 +1 · · · xm0 −1 6= 0,
from which it follows that the product
xi0 xi0 +1 · · · xm0 −1 zσ(0) zσ(1) · · · zσ(i0 −1) 6= 0.9
Now, since xi0 ∈ J, and by our definition of n0 ∈ N and z0 , z1 , . . . , zn0 ∈ J above, it follows
that there is some n ∈ {0, 1, . . . , n0 − 1} such that
zn xi0 +1 · · · xm0 −1 zσ(0) zσ(1) · · · zσ(i0 −1) 6= 0,
from which it follows that
zσ(0) zσ(1) · · · zσ(i0 −1) zn xi0 +1 · · · xm0 −1 6= 0,
=(i0 +1)
from which it follows that there is some τ ∈ n0
that
Y
Y
zτ (j) ·
j<i0 +1
properly extending σ, τ = σn, such
xj 6= 0,
i0 +1≤j<m0
0
contradicting our definition of i0 . Hence, i0 = m0 and there is a string τ ∈ n=m
such that
0
Y
zτ (j) 6= 0,
j<m0
9Here
we are using the commutativity of R.
12
CHRIS J. CONIDIS
contradicting our definition of m0 . Hence no such sequence x0 , x1 , . . . , xm0 −1 ∈ J exists and
J m0 = 0.
In the next section we will use the author’s results in [Con10] to prove that NIL0 implies
WKL0 over RCA0 . We now use the previous theorem (i.e. Theorem 3.1) to show that WKL0 +
IΣ2 implies ART0 over RCA0 .
Theorem 3.2. WKL0 + IΣ2 implies ART0 over RCA0 .
Proof. The proof is very similar to the part of the proof of [Con10, Theorem 3.4] given in
[Con10, Section 3.5]. We assume that the reader is familiar with [Con10, Section 3.5] and
therefore we will only give a sketch of the proof and leave the details (which can essentially
be found in [Con10, Section 3.5]) to the reader. Let R be a ring with an infinite strictly
ascending chain of ideals I0 ⊂ I1 ⊂ I2 ⊂ · · · and a corresponding infinite sequence of
elements {xk }k∈N such that for all i ∈ N we have that xi ∈ Ii+1 \ Ii . We will construct, via
WKL0 + IΣ2 , an infinite strictly descending chain of ideals J0 ⊃ J1 ⊃ J2 ⊃ · · · in R.
First of all, via [Con10, Section 3.5.1] we can assume that R has only finitely many maximal
ideals, M0 , M1 , . . . , Mn0 −1 ⊂ R, n0 ∈ N (otherwise we can construct an infinite strictly
descending chain of ideals in R by repeatedly intersecting maximal ideals; see [Con10, Section
3.5.1] for more details). In this case, we have that the Jacobson radical of R, J ⊂ R, is equal
to the product M0 M1 · · · Mn0 −1 (see [Con10, Section 3.5.2] for more details), and by Theorem
3.1 above it follows that there is a number m0 ∈ N such that J m0 = (M0 M1 · · · Mn0 −1 )m0 = 0.
Now, for all 0 < i ≤ n0 m0 , write i = ki n0 + ri , ri , ki ∈ N, ri < n0 , and define
Ai = J ki M0 M1 · · · Mri .
Although we cannot actually prove that Ai , 0 < i ≤ n0 m0 , exist via WKL0 , we have that
Ai is Σ01 -definable for all 0 < i ≤ n0 m0 , and we can use IΣ2 (as in [Con10, Section 3.5.2])
to find the largest number i0 ∈ N, i0 ≤ n0 m0 , such that there exist infinitely many k ∈ N
such that xk ∈ Ai0 . It follows that there is a computably enumerable subset of {xk }k∈N that
live inside Ai0 \ Ai0 +1 . Let f : N → N be a total computable strictly increasing function
such that xf (k) ∈ Ai0 \ Ai0 +1 for all k ∈ N. Now, note that the quotient Ai0 /Ai0 +1 is an
R/Mri0 +1 -vector space and by our construction of {xk }k∈N above it follows that for all k ∈ N
xf (k) is not an R-linear combination of {xf (j) }j>k . Now, by an argument similar to the one
given in [Con10, Section 3.5.2] we can use WKL0 to construct an infinite strictly descending
chain of ideals J0 ⊃ J1 ⊃ J2 ⊃ · · · such that for all k ∈ N we have that xf (k) ∈ Jk \ Jk+1 . In the next section we will use the author’s results in [Con10] to prove that ART0 implies
WKL0 over RCA0 + IΣ2 . We now use Theorem 3.1 above to show that WKL0 implies the
structure theorem for Artinian rings (i.e. ARTs0 ) over RCA0 . First, however, we need to prove
the Chinese Remainder Theorem (CRT0 ) in RCA0 . We now state the Chinese Remainder
Theorem.
Theorem 3.3 (Chinese Remainder Theorem (CRT0 )). Let n ∈ N, n ≥ 2, and R be a
commutative ring with ideals A1 , A2 , . . . , An ⊂ R such that for all 1 ≤ i < j ≤ n we have
that Ai + Aj = R. Then the map
ϕ : R → R/A1 × R/A2 × · · · × R/An
is a surjection with kernel
A1 A2 · · · An = A1 ∩ A2 ∩ · · · ∩ An .
Lemma 3.4. RCA0 implies CRT0 .
Proof. The following proof of the Chinese Remainder Theorem in RCA0 is similar to many
of the proofs given in standard algebra texts, and does not use induction.
First of all, by the proof of [Con10, Proposition 2.16] (which is the same as the classical
proof of the same fact), it follows that the kernel of ϕ is as claimed above.
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
13
Suppose now that n = 2 in the statement of CRT0 above. Then there exist a1 ∈ A1 and
a2 ∈ A2 such that a1 + a2 = 1 ∈ R. Then a1 x2 + a2 x1 ∈ R maps to (x1 , x2 ) ∈ R/A1 × R/A2
under ϕ, which shows that ϕ is surjective.
Now, let n ∈ N (be not necessarily 2). For each 2 ≤ i ≤ n there exist elements ai ∈ A1
and bi ∈ Ai such that
ai + bi = 1 ∈ R.
Q
Q
Furthermore, the product 2≤i≤n (ai + bi ) = 1R and lies in A1 + 2≤i≤n Ai , and hence
Y
A1 +
Ai = R.
2≤i≤n
Q
Note that the product of ideals 2≤i≤n Ai need not exist in RCA0 . We can now apply our
proof of the Chinese Remainder Theorem in the case n = 2 to obtain an element y1 ∈ R
such that
y1 ≡ 1 mod A1 , y1 ≡ 0 mod Aj , 2 ≤ j ≤ n.
Similarly, there exist elements y2 , . . . , yn ∈ R such that for all 2 ≤ i ≤ n we have that
yi ≡ 1 mod Ai , yi ≡ 0 mod Aj , 1 ≤ j ≤ n, j 6= i.
P
It follows that the element x = 1≤i≤n xi yi maps to
(x1 , x2 , . . . , xn ) ∈ R/A1 × R/A2 × · · · × R/An
under ϕ, which shows that ϕ is surjective. This completes the proof of the lemma.
We will now use the fact that CRT0 holds in RCA0 to prove the structure theorem for
Artinian rings in WKL0 .
Theorem 3.5. WKL0 implies ARTs0 over RCA0 .
Proof. We reason in WKL0 , and assume that the reader is familiar with the proof of [Con10,
Theorem 3.4]. Let R be an Artinian ring. Let M0 , M1 , . . . , Mn0 −1 ⊂ R, n0 ∈ N, n0 ≥ 2,
be the maximal ideals of R (see [Con10, Section 3.5.1] or the proof Theorem 3.2 above for
more details). Note that if n0 = 1 then R is local and the theorem follows trivially. For all
0 ≤ j < n0 let
[
xj ∈ Mj \
Mk
0≤k<n0
k6=j
and let J = M0 ∩ M1 ∩ · · · ∩ Mn0 −1 = M0 M1 · · · Mn0 −1 ⊂ R be the Jacobson radical of R. By
[Con10, Theorem 3.4] it follows that M0 , M1 , . . . , Mn0 −1 are also the prime ideals of R and J is
also the nilradical of R. Via Theorem 3.1 above and Π01 -induction, let hm0 , m1 , . . . , mn0 −1 i ∈
N be the least n0 −tuple such that
m
0 −1
M0m0 M1m1 · · · Mn0n−1
= 0.
m
Now, we claim that for every 0 ≤ j < n0 the ideal Mj j exists. To see why, without loss
of generality assume that j = 0 (the general argument is similar). First, we claim that there
exists
Y
x∈
Mimi \ M0 .
Otherwise, we would have that
Q
1≤i<n0
mi
⊆
1≤i<n0 Mi
x=
M0 and hence the element
Y
i
xm
i
1≤i<n0
would satisfy x ∈ M0 , from which it follows thatQsome xi , 1 ≤ i < n0 , is in M0 (since
M0 is a prime ideal), a contradiction. Hence x ∈ 1≤i<n0 Mimi \ M0 as claimed. Next, we
claim that Ann(x) = M0n0 , from which it follows that M0n0 exists. By our construction of
m0 , m1 , . . . , mn0 −1 ∈ N it is easy to see that M0m0 ⊆ Ann(x). Now, suppose that there is
14
CHRIS J. CONIDIS
some y ∈ Ann(x) \ M0n0 and use WKL0 to construct an ideal I such that M0n0 ⊆ I ⊂ Ann(x)
and y ∈
/ I. Consider the ring R/I. Since R/I is the quotient of an Artinian ring, RCA0
proves that R/I is Artinian and every ideal in R/I corresponds to an ideal in R containing
I via pullback. We claim that R/I is a local ring with unique maximal ideal M 0 ⊂ R/I.
For suppose that there is another maximal ideal M 6= M 0 in R/I, then M must correspond
to a maximal ideal M in R containing I. But then we have that
M ⊃ I ⊇ M0m0
and so it follows that M contains M0 , from which it follows that M = M0 , since M, M0 are
maximal and hence prime ideals in R. Therefore, we have that M = M 0 , a contradiction.
Now,
since R/I is a local Artinian ring with unique maximal ideal M 0 it follows that x ∈
Q
mi
1≤i<n0 M i \ M 0 is a unit in R/I. But, on the other hand we have that y 6= 0 in R/I
(by our construction of I above) and (by our construction of y above) y · x = 0 in R/I, a
contradiction. Hence, no such y ∈ Ann(x)\M0m0 exists, and thus Ann(x) = M0m0 as claimed.
m
For all 0 ≤ i < j < n0 it follows that Mimi + Mj j = R. Otherwise we could use WKL0
to construct an ideal I such that Mim0 + Mjm0 ⊆ I ⊆ Mk ⊂ R, 0 ≤ k < n0 , a contradiction
since Mk is a prime ideal and so xi , xj ∈ Mk . Furthermore, by an argument similar to one
given in the last half of the previous paragraph it follows that for every 0 ≤ i < n0 the ring
R/Mimi is a local ring with unique maximal ideal M i ⊂ R/Mimi . Now, we can apply the
Chinese Remainder Theorem (CRT0 ) to construct a homomorphism
m
0 −1
ϕ : R → R/M0m0 × R/M1m1 × · · · × R/Mn0n−1
given by
ϕ(x) = (xR/M0m0 , xR/M1m1 , . . . , xR/M mn0 −1 )
n0 −1
with kernel
m
0 −1
M0m0 M1m1 · · · Mn0n−1
= 0.
Hence, ϕ is an isomorphism and the theorem follows.
3.1. A second proof of ART0 via WKL0 + IΣ2 . We now give a second (different) proof of
Theorem 3.2 above. We assume that the reader is familiar with the proofs of Theorems 3.1
and 3.2 above.
A different proof of Theorem 3.2. We reason in WKL0 + IΣ2 . First, let R be a local ring with
unique maximal ideal M ⊂ R and an infinite strictly ascending chain of ideals
I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ M ⊂ R.
We aim to show that R has an infinite strictly descending chain of ideals J0 ⊃ J1 ⊃ J2 ⊃ · · · ;
this suffices to prove the theorem. Let z0 , z1 , . . . , zn0 −1 ∈ M ⊂ R, T ⊂ n<N
0 , and m0 ∈ N,
be as in the proof of Theorem 3.1 above, and let {xk }k∈N be as in the proof of Theorem
3.2 above. Without loss of generality we can assume that T is a finite tree; otherwise the
theorem follows as in Theorem 3.1 above. So far our proof has been very similar to that of
Theorem 3.2 above, but here is where they start to diverge.
First of all, let
U = R \ M,
and note that U is the set of units in R since R is a local Artinian ring, and we are reasoning
in WKL0 (see [Con10, Theorem 3.4] for more details). Now, via IΣ2 let l0 ∈ N, l0 < m0 , be
the least number such that there exist infinitely many k ∈ N and u0,k , u1,k , . . . , uk−1,k ∈ U =
0
R \ M such that for all σ ∈ n≥l
∩ T we have that
0
y k = xk +
k−1
X
j=0
uj,k xj ∈ Ann(zσ ),
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
15
where σ ∈ n<N
0 ,
zσ =
Y
zσ(i) ∈ M ⊂ R,
i<|σ|
and z∅ = 1R . Note that l0 > 0 since z∅ = 1R . Let σ0 ∈ T , |σ0 | = l0 − 1, be such that there
do not exist infinitely many k ∈ N and u0,k , u1,k , . . . , uk−1,k ∈ U as above (i.e. let σ0 ∈ T be
a witness to the fact that l0 is minimal). Finally, via our construction of σ0 and RCA0 , let
{yk }k∈N be an infinite sequence of elements of R as displayed above where σ is any successor
of σ0 in n<N
0 .
We claim that there exists k0 ∈ N such that for all k ≥ k0 , k ∈ N, we have that zσ0 yk
is not an R-linear combination of {zσ0 yl }l>k . To prove our claim suppose otherwise (for a
contradiction). In this case by hypothesis we have that for all k ∈ N there exists j > k,
j ∈ N, such that zσ0 yj is an R-linear combination of {zσ0 yl }l>j . Now, first of all note that
by our constructions of σ0 ∈ T and zσ0 , z0 , . . . , zn0 −1 ∈ M ⊂ R above it follows that for all
k ∈ N and x ∈ M we have that
x · zσ0 yk = 0
since, by our construction of {yk }k∈N in the previous paragraph, for all k ∈ N and successor
strings σ ⊃ σ0 , σ ∈ n<N
0 , we have that zσ yk = 0. Hence, under our current hypothesis and
our remarks in the previous sentence, we can assume that there exist infinitely many j ∈ N
such that zσ0 yj is a U -linear combination of {zσ0 yl }l>j . Now, since U = R \ M is the set of
units in the local ring R, every equation of the form
zσ0 yj = zσ0 ·
l1
X
ul yl , j ∈ N, l1 > j, ul ∈ U,
l=j+1
can be manipulated/rearranged (via division by ul1 ∈ U ) to read
!
X
zσ0 yl0 +
uj yj = 0.
j<l0
The fact that this can be done for unboundedly many (i.e. arbitrarily large) l0 , j ∈ N contradicts our construction of σ0 in the previous paragraph. Therefore, the claim we made in
the first sentence of this paragraph holds, and we can use this claim along with WKL0 to
construct an infinite descending chain of ideals J0 ⊃ J1 ⊃ J2 ⊃ · · · in R such that for all
k ∈ N we have that zσ0 yk0 +k ∈ Jk \ Jk+1 . This completes the proof of the theorem in the
case when R is a local ring. We now turn our attention to proving the theorem when R is
not a local ring.
Assume that R is an Artinian ring, with finitely many maximal ideals M0 , M1 , . . . , Mm0 −1 ⊂
R, m0 ∈ N, and an infinite strictly ascending chain of ideals I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ R. The
fact that R has finitely many maximal ideals is a consequence of [Con10, Theorem 3.4]. Let
x0 , x1 , x2 , . . . ∈ R, xi ∈ Ii+1 \ Ii , i ∈ N, be as before. Note that for every n ∈ N there
exists 0 ≤ i < m0 such that (xn : In ) ⊆ Mi , since otherwise (using our previous arguments
in [Con10, Theorem 3.4]) we could use this fact along with WKL0 to construct a maximal
ideal M ⊂ R that is different from M0 , M1 , . . . , Mm0 −1 , a contradiction. For all n ∈ N, let
in : N → N, be a uniformly computable sequence of total functions such that the range of in
is (xn+1 : In ) ⊂ R, and for all n, k ∈ N let
In,k = {in (0), in (1), . . . , in (k)}
and note that {In,k }n,k∈N is a uniformly computable listing of finite sets. Now, construct the
computable tree T1 ⊆ m<N
such that for all σ ∈ m<N
we have that
0
0
σ ∈ T1 ⇔ (∀τ ⊆ σ)[I|τ |,|σ| ⊆ Mσ(|τ |−1) ].
16
CHRIS J. CONIDIS
It follows from bounded Π01 -comprehension and our previous remarks in this paragraph that
for every n ∈ N there is a string of length n on T1 . It is also not difficult to verify that if
f ∈ mN0 is an infinite path through T1 , then for every n ∈ N we have that
(xn : In ) ⊆ Mf (n) .
But, by WKL0 it follows that there is an infinite path f ∈ mN0 through T1 ⊆ m<N
0 , since
T1 is an infinite computable tree. Furthermore, it follows from IΣ2 that there is a number
i0 ∈ N, 0 ≤ i0 < m0 , such that there exist infinitely many n ∈ N such that f (n) = i0 . Let i0
be as in the previous sentence and use RCA0 to construct
P = {n ∈ N : f (n) = i0 }.
It follows that P is an infinite set. Let
P = {p0 < p1 < · · · < pk < · · · , k ∈ N}
be a listing of the (infinitely many) elements of P . Let
R0 = RMi0 = R[(R \ Mi0 )−1 ]
be the localization of R at the prime ideal Mi0 (RCA0 suffices to construct R0 ). By our
construction of the infinite set/sequence P = {pk }k∈N and Lemma 2.1 in Subsection 2.3
above it follows that the infinite ascending chain of ideals
Ip0 ⊂ Ip1 ⊂ Ip2 ⊂ · · · ⊂ Ipk ⊂ · · · , k ∈ N,
in R corresponds to an infinite strictly ascending chain of ideals in R0 . Now we can apply our
previous proof (for local rings) to the local ring R0 to produce an infinite strictly descending
chain of ideals J00 ⊃ J10 ⊃ J20 ⊃ · · · in R0 corresponding to an infinite strictly descending
chain of ideals J0 ⊃ J1 ⊃ J2 ⊃ · · · in R. This ends the proof of the theorem.
4. Deriving weak König’s lemma from NIL0 , ART0 , and ARTs0
The main purpose of this section is to establish optimal lower bounds on the reverse
mathematical strengths of NIL0 , ART0 , and ARTs0 . More specifically, we will prove that
(1) NIL0 implies WKL0 over RCA0 ;
(2) ART0 implies WKL0 over RCA0 ; and
(3) ARTs0 implies WKL0 over RCA0 .
The author already proved (2) in [Con10, Theorem 4.1]. Our proof of (1) is based on the same
construction (i.e. [Con10, Theorem 4.1]). Our proof of (3) is based on [DLM07, Theorem
3.2, Proposition 3.4].
Theorem 4.1. NIL0 implies WKL0 over RCA0 .
Proof. We reason in RCA0 . Let R be as in [Con10, Theorem 4.1]. Recall that R is an integral
domain such that the only computable ideals in R form an infinite strictly ascending chain
{0} = I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ I∞ ⊂ R, where I∞ = ∪n∈N Ii , and every other ideal of R,
J 6= Ip , R, p ∈ N ∪ {∞}, is of PA Turing degree (for more information see Section 2.3 above
or [Con10, Theorem 4.1]). Now, since R is an integral domain, it follows from NIL0 that
the intersection of the maximal ideals of R must be zero. It follows that R must contain a
maximal ideal M that is not equal to any Ip , p ∈ N ∪ {∞}, and therefore M must be of PA
Turing degree. Relativizing this argument to every oracle A ⊂ N proves that NIL0 implies
WKL0 over RCA0 .
Theorem 4.2. ARTs0 implies WKL0 over RCA0 .
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
17
Proof. Our proof uses some elements of the proof of [DLM07, Theorem 3.2], but is also
somewhat different. First, we will construct a computable ring R using some of the ideas
found in [DLM07, Theorem 3.2], then we will note that our construction relativizes to an
arbitrary oracle X ⊂ N to produce a ring RX with various special properties having to do
with PA Turing degrees relative to X. Afterwards, we will show that there is no model of
RCA0 +¬WKL0 +ARTs0 by assuming that such a model M exists, and deriving a contradiction
based on an application of the axiom ARTs0 to the ring RX , where X ⊆ N is an oracle in
M chosen so that there is no PA Turing degree relative to X in M (note that the oracle X
exists because M satisfies ¬WKL0 by hypothesis). More precisely, we will apply ARTs0 to RX
to deduce the existence of a PA Turing degree relative to X, which is a contradiction by the
way we chose X.
We now construct R = R∅ . First, however, let C0 , C1 ⊂ N be disjoint computably enumerable sets such that any separator for C0 , C1 is of PA Turing degree. Let c0 , c1 : N → N
be one-to-one total computable functions such that the range of ci is Ci , i ∈ {0, 1}. Let
Z = {z0 < z1 < z2 < · · · < zi < · · · , i ∈ N} ⊂ N
be an infinite computable set disjoint from both C0 and C1 , and for all k ∈ N let
z0 (k) = zc0 (k) ∈ Z and z1 (k) = zc1 (k) ∈ Z.
It is well-known that Z exists; see [KS07, Lemma 2.6], for example. Recall that
Z∞ = Z[X0 , X1 , X2 , . . .]
is the free polynomial ring over Z with infinitely many indeterminates X0 , X1 , X2 , . . ., that
F is the field of fractions of Z∞ , and that Zk = Z[X0 , X1 , . . . , Xk ]. For all p ∈ Z∞ define
s(p) ∈ N to be the least number s0 such that p ∈ hXc0 (k) : k ∈ N, 0 ≤ k ≤ s0 iZ∞ ; let
s(p) = ∞ if no such s0 exists. Also let mp ∈ N, p ∈ Z∞ , be the least m ∈ N such that
p ∈ Zm . Finally, let
Xc0 (k)
D0 =
: k ∈ N, p ∈ Z∞ , k < s(p) ,
p
Xz0 (k)
D1 =
: k ∈ N, p ∈ Z∞ , z0 (k) > s(p) ,
p
Xc0 (s(p)) − Xz1 (k)
D2 =
: k ∈ N, p ∈ Z∞ , s(p) < ∞, mp < z1 (k) ,
p
and
1
D3 =
:k∈N .
Xc1 (k)
Note that D0 , D1 , D2 , and D3 are c.e. subsets of the computable field F . It follows that the
subring generated by
3
[
R0 = Z∞ ∪
Di
i=0
in F is computably isomorphic to a computable ring R. Without any loss of generality we
identify R with R0 ⊂ F . Note that R0 ∼
= R are integral domains.
For each n ∈ N, let
An = {c0 (0), c0 (1), . . . , c0 (n)} ∪ Z ⊂ N.
∼
By our construction of R = R0 above it follows that for every n ∈ N the ideal
In = hXk : k ∈ An iR ⊂ R
is computable, and therefore exists via RCA0 . Let
[
I∞ =
In .
n∈N
18
CHRIS J. CONIDIS
Now, by our construction of R above, it follows that for all n ∈ N we have that Xc0 (n+1) ∈
/ In
since (by our construction of R above) for all n ∈ N we have that the numerator of every
element of In ⊂ R ∼
= R0 is in hXk : k ∈ An iZ∞ . We leave the easy verification of these facts
to the reader.
Now, we claim that R is an Artinian ring unless there is a PA Turing degree. To prove
this claim, first of all note that if I is a nontrivial ideal such that I * I∞ then, by our
construction of D0 and D3 above, it follows that I has PA Turing degree since we have that
(a) Xc0 (k) ∈ I, for all k ∈ N; and
(b) Xc1 (k) ∈
/ I, for all k ∈ N.10
A similar (simpler) argument applies in the case I = I∞ . Hence, in a model of RCA0 +¬WKL0
we do not have any such ideals I. Now, by our construction of D0 above and our previous
remarks in this paragraph it follows that in any model of RCA0 + ¬WKL0 every nontrivial
ideal of R is contained in IN , for some N ∈ N. Let
J0 ⊃ J1 ⊃ J2 ⊃ · · ·
be an infinite strictly descending chain of ideals in R, and, via Π01 -induction, let n0 ∈ N be
maximal such that In0 ⊂ Jk , for all k ∈ N. Note that if n0 does not exist, then it follows
that I∞ ⊂ Jk , for all k ∈ N, from which it follows (from our previous remarks) that Jk is of
PA Turing degree for all k ∈ N. Therefore, our claim is valid in the case when n0 does not
exist. On the other hand, if n0 ∈ N exists, then let m0 ∈ N be such that for all m ≥ m0 we
have that In0 +1 is not contained in Jm . It follows that
(1) In0 +1 ⊇ Jm0 ⊃ Jm0 +1 ⊇ In0 ; and
(2) Xc0 (n0 +1) ∈
/ Jm0 .
Now, because of (1) and (2) above, as well as our constructions of D1 and D2 (also above),
it follows that we have
(a) Xz0 (k) ∈ Jm0 , for almost all k ∈ N; and
(b) Xz1 (k) ∈
/ Jm0 , for almost all k ∈ N.
By our constructions of Z ⊂ N and z0 , z1 : N → N above it follows that Jm0 is of PA Turing
degree, which proves the claim we made in the first sentence of this paragraph above. Note
that all of our constructions and arguments thus far can be relativized to any given oracle
X ⊆ N. In other words, for any given oracle X ⊆ N there is an X−computable integral
domain RX such that RX is an Artinian ring unless there exists a PA Turing degree relative
to X.
Now, suppose for a contradiction that ARTs0 does not imply WKL0 over RCA0 and let M
be any model of RCA0 + ¬WKL0 + ARTs0 , and let X ∈ M be a subset of the universe of M
such that M does not contain a PA Turing degree relative to X. Throughout this paragraph
we will work within the model M. Let RX ∈ M be the X−computable integral domain
described in the final sentence of the previous paragraph (RX exists in M via RCA0 ). Now,
by our construction of RX and our construction of X ∈ M via our hypothesis ¬WKL0 , it
follows that RX is an Artinian integral domain in M. Hence, we can apply our hypothesis
ARTs0 to RX to conclude that RX is isomorphic to a finite direct product of local Artinian
rings, i.e. there exists k0 ∈ N and local Artinian rings R0,X , R1,X , . . . , Rk0 ,X with respective
unique maximal ideals M0,X , M1,X , . . . , Mk0 ,X , Mi,X ⊂ Ri,X , 0 ≤ i ≤ k0 , such that
RX ∼
= R0,X × R1,X × · · · × Rk ,X = ZX
0
via a given isomorphism ϕ : RX → ZX . But, since RX is an integral domain it follows that
k0 = 0 since otherwise RX contains nontrivial zero divisors. Hence, RX is a local Artinian
integral domain with unique maximal ideal MX ⊂ RX , MX , RX ∈ M. But, by our previous
remarks and constructions, and the maximality of MX , it follows that either MX is not
10Note
k ∈ N.
that by our construction of D1 and R we have that Xc1 (k) ∈
/ B for any proper ideal B ⊂ R and
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
19
contained in I∞ , or else MX = I∞ . But in either case (again, by previous remarks) it follows
that MX is of PA Turing degree relative to X, a contradiction. Therefore we must have that
ARTs0 implies WKL0 over RCA0 .
5. Coding the Halting Set into two-sided descending chains in
noncommutative rings
We end our paper with a definability theorem that says that certain generalizations of
Theorems 3.1 and 3.2 above are not true in the context of noncommutative rings. However,
this theorem has no reverse mathematical counterpart because the statement “every Artinian
ring is Noetherian” is not true in the context of noncommutative rings and two-sided ideals.
The following interesting question is still open.
Question 5.1. What are the reverse mathematical strengths of NILn0 and ARTn0 over RCA0 ?
We make the following conjecture.
Conjecture 5.2. NILn0 and ARTn0 are both equivalent to ACA0 over RCA0 .
We now prove the final theorem of this article, which is interesting in light of Theorems 3.1
and 3.2 above because Theorems 3.1 and 3.2 say that the analog of the following theorem for
commutative rings is not true. For simplicity, we prove the following theorem in the context
of ω-models.
Theorem 5.3. Let M be an ω-model of RCA0 . Then M contains a local noncommutative
ring R with unique (computable) maximal ideal M ⊂ R such that:
(1) R contains a uniformly computable strictly ascending chain of ideals
{0} = I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ M ⊂ R;
(2)
(3)
(4)
(5)
M is not nilpotent, i.e. for all n ∈ N, M n 6= 0;
Every element of R \ M is a unit (i.e. has both left and right inverses);
R contains an infinite strictly descending chain of ideals (not necessarily computable);
Every infinite strictly descending chain of ideals in R, J0 = R ⊃ J1 ⊃ J2 ⊃ · · · ,
computes the Halting Set ∅0 (i.e. we have that J = ⊕k∈N Jk computes ∅0 ).
Proof. We work exclusively within the given ω-model (of RCA0 ) M. Recall that ω-models
of RCA0 are essentially Turing ideals, and satisfy full induction. Throughout this proof we
will identify N with the universe of M. In other words, we will have that
N = ω = {0, 1, 2, . . .}.
Furthermore, R ∈ M will be a computably enumerable subring of a larger noncommutative
ring Q that we construct first. We will construct Q as the union of increasing subrings,
Q = ∪m∈N Qm , Qm ⊂ Qm+1 . First, we let X = {Xn : n ∈ N} and define Q0 = Q[X]/I0 ,
where I0 ⊂ Q[X] is the two-sided ideal generated by products of the form {Xi Xj : i, j ∈ N}.
By construction it is clear that Q0 is a computable noncommutative ring with identity, and
every element of Q0 has a unique representative of the form
q = q0 +
k
X
qi Xi−1 ,
i=1
for some q0 , q1 , . . . , qk ∈ Q and k ∈ N, k > 0. It follows that the nonzero monomial elements
of Q0 are Q-multiples of Xi , for some i ∈ N. For all i ∈ N define the index of the monomial/indeterminate Xi via ind(Xi ) = {i} ⊂ N.
Constructing Q1 ⊃ Q0 :
20
CHRIS J. CONIDIS
Next, let M0 ⊂ Q[X0 ] be the set of monomials in Q[X0 ] with coefficient 1 that correspond
to nonzero elements of Q0 = Q[X0 ]/I0 (for now these are precisely the indeterminates Xn ,
n ∈ N), and define
Y01 = {YBA : A, B ∈ M0 , ind(A) > ind(B)}, Z01 = {ZBA : A, B ∈ M0 , ind(A) > ind(B)},
X1 = Y01 ∪ Z01 .
For all WBA ∈ X1 , W ∈ {Y, Z}, A, B ∈ M0 , ind(A) > ind(B), define ind(WBA ) = ind(A) ∪
ind(B). We will define Q1 as a quotient of Q[X0 ∪ X1 ]. Now, define
I10 = {YBA AZBA − B : A, B ∈ M0 , ind(A) > ind(B)},
and let
Q01 = Q[X0 ∪ X1 ]/hI0 , I10 iQ[X0 ∪X1 ] .
We claim that Q01 is a computable ring; proving so amounts to showing that hI0 , I10 iQ[X0 ∪X1 ]
is a computable ideal of Q[X0 ∪ X1 ]. Note that the generating set I10 corresponds to multiplication rules of the form:
(∗) YBA AZBA = B, A, B ∈ M0 , ind(A) > ind(B)
in the quotient Q01 = Q[X0 ∪ X1 ]/hI0 , I10 i. We call a monomial m ∈ Q[X0 ∪ X1 ] reduced
whenever m contains no factors of the form YBA AZBA as in the multiplication rule (∗) above.
Let red(Q[X0 ∪ X1 ]) be the set of reduced monomials in Q[X0 ∪ X1 ]. Via substitutions corresponding to the multiplication rules described above, it follows easily that every monomial m
of Q01 has a representative in red(Q[X0 ∪ X1 ]) and that such a representative can be obtained
uniformly and effectively from m.
Now, we claim that every monomial m ∈ Q01 = Q[X0 ∪ X1 ]/I10 , m ∈ Q[X0 ∪ X1 ], has a
unique representative in red(Q[X0 ∪ X1 ]). Let m ∈ Q[X0 ∪ X1 ] be a monomial. First we
prove the subclaim that says for all m0 ∈ red(Q[X0 ∪ X1 ]) such that there exists a sequence
of monomials
m = m0 , m1 , m2 , . . . , mk = m0 ∈ Q[X0 ∪ X1 ]
such that k ∈ N is minimal and for all 0 ≤ i < k, mi+1 is obtained from mi via an
application of a single multiplication rule (∗) described in the previous paragraph, then for
all 0 ≤ i < k, mi+1 was obtained from mi by substituting B ∈ M0 for some product of
the form YBA AZBA , A ∈ M0 , ind(A) > ind(B). In other words, the degree of the monomials
m = m0 , m1 , m2 , . . . , mk = m0 ∈ Q[X0 ∪ X1 ] is strictly decreasing. For suppose (for a
contradiction) that we have deg(mi ) < deg(mi+1 ). In other words, suppose that there exists
0 ≤ i < k such that mi+1 is obtained by replacing some B ∈ M0 with a product of the form
YBA AZBA , A ∈ M0 , ind(A) > ind(B) via some multiplication rule (∗) above. Furthermore,
(via ∆01 -comprehension) let i0 ∈ N be the largest 0 ≤ i < k with this property, and let
B0 , A0 ∈ M0 , ind(A0 ) > ind(B0 ), be such that mi0 +1 is obtained from mi0 by replacing B0
with YBA00 A0 ZBA00 . Now, by definition of i0 and our multiplication rules and ∆01 -induction it
follows that every mi , i0 + 1 ≤ i < k, has a factor of the form YBA00 A0 ZBA00 unless we replaced
it by B0 via the unique multiplication rule of type (∗) above that allows us to do so. Now
let
mi0 = ai0 YBA00 A0 ZBA00 bi0 , mi0 +1 = ai0 +1 YBA00 A0 ZBA00 bi0 +1 , mi0 +2 = ai0 +2 YBA00 A0 ZBA00 bi0 +2 , . . .
. . . ml = al B0 bl ,
for some i0 < l ≤ k and monomials ai0 = ai0 +1 , bi0 = bi0 +1 , ai0 +2 , bi0 +2 , . . . , al , bl ∈ Q[X0 ∪ X1 ]
such that for all i0 + 1 < i < l exactly one of ai , bi is obtained from the corresponding
ai−1 , bi−1 via some multiplication rule (∗) above. For all i0 + 1 ≤ i < l let m̂i ∈ Q[X0 ∪ X1 ] be
the monomial obtained from mi by replacing the distinguished factor YBA00 A0 ZBA00 displayed
above by the indeterminate B0 . Then it follows that the sequence
m = m0 , m1 , m2 , . . . , mi0 , m̂i0 +2 , . . . , m̂l−1 , ml+1 , . . . , mk = m0
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
21
witnesses the non-minimality of k, a contradiction. Now, note that for all monomials m ∈
Q[X0 ∪ X1 ] and m0 ∈ red(Q[X0 ∪ X1 ]) as above (in this paragraph) Σ01 -induction implies the
existence of a minimal k ∈ N. This proves the subclaim. Our initial claim follows from our
subclaim and the fact that “multiplication rules (∗) do not overlap.” In other words, for
any given A, B ∈ M0 there is at most one multiplication rule (∗) that mentions both A and
B. Moreover, it follows that to obtain m0 from m one simply substitutes every occurrence
of a factor of the form YBA AZBA , A, B ∈ M0 , ind(A) > ind(B), by B, and this process is
well-defined by our comments in the previous two sentences. It follows that Q[X0 ∪ X1 ]/I10
is a computable ring.
Now, if m ∈ Q[X0 ∪ X1 ] is a monomial and m0 ∈ Q[X0 ∪ X1 ] is a monomial obtained from
m via a single application of a multiplication rule (∗) above such that deg(m) > deg(m0 )
then it easily follows that m0 ∈ hI0 iQ[X0 ∪X1 ] (i.e. m0 contains a factor of the form Xi Xj ,
i, j ∈ N,) whenever m ∈ hI0 iQ[X0 ∪X1 ] (i.e. m contains a factor of the form Xi Xj , i, j ∈ N).
By our results in the previous paragraph it follows that Q01 = Q[X0 ∪ X1 ]/hI0 , I10 iQ[X0 ∪X1 ]
is a computable ring, and moreover every monomial m ∈ Q01 , m ∈ Q[X0 ∪ X1 ], has a
unique representative in red(Q[X0 ∪ X1 ]). Furthermore, it follows that every element p ∈ Q01 ,
p ∈ Q[X0 ∪ X1 ], has a unique representative as a sum of monomials in red(Q[X0 ∪ X1 ]).
Finally, note that Q0 embeds into Q01 , i.e. Q0 ,→ Q01 , via the natural map induced by the
natural embedding Q[X0 ] ,→ Q[X0 ∪ X1 ]. This follows from the fact that every indeterminate
YBA ∈ Y01 ,ZBA ∈ Z01 , A, B ∈ M0 , ind(A) > ind(B), appears in exactly one element of the
generating set I10 ⊂ Q[X0 ∪ X1 ], and also every generator in I10 has an appearance of some
YBA ∈ Y01 and also some ZBA ∈ Z01 .
We now define Q1 as a quotient of the form Q1 = Q01 /I1 , for some ideal I1 ⊂ Q01 . Let I1
be the ideal of Q01 generated by the images m of monomials m ∈ red(Q[X0 ∪ X1 ]) in Q01 such
that m contains a factor of the form
Y Z, Y ∈ Y01 ⊂ X1 , Z ∈ Z01 ⊂ X1 ; or
ZY , Y ∈ Y01 ⊂ X1 , Z ∈ Z01 ⊂ X1 ; or
XYBA , A, B, X ∈ M0 , ind(A) > ind(B); or
ZBA X, A, B, X ∈ M0 , ind(A) > ind(B); or
YBA X, A, B, X ∈ M0 , ind(A) > ind(B), X 6= A; or
XZBA , A, B, X ∈ M0 , ind(A) > ind(B), X 6= A; or
C
, A, B, C, D, X ∈ M0 , ind(A) > ind(B), ind(C) > ind(D), such that either
YBA XZD
A 6= C or B 6= D or X 6= A; or
(viii) YBA YDC , A, B, C, D ∈ M0 , ind(A) > ind(B), ind(C) > ind(D), A 6= D; or else
C
(ix) ZBA ZD
, A, B, C, D ∈ M0 , ind(A) > ind(B), ind(C) > ind(D), B 6= C.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
This completes our construction of Q1 . We now show that Q1 is a computable ring - i.e. we
prove that the ideal I1 ⊆ Q01 generated by the monomials with factors given in (i)-(ix) above
is in fact computable. Note that Q0 ,→ Q1 since Q0 ,→ Q01 and every monomial generator
(i)-(ix) above contains a factor in X1 .
To show that I1 ⊂ Q01 is computable it suffices to show that if m0 , m1 ∈ Q01 , m0 , m1 ∈
red(Q[X0 ∪ X1 ]), are monomials such that m0 is a generator of I1 (defined in the previous
paragraph) then both of the products m1 ·Q01 m0 and m0 ·Q01 m1 are also generators of I1 ⊆ Q01 .
But this follows directly from our construction of (i)-(ix) in the previous paragraph and our
construction of Q01 above. More precisely, by our construction of the multiplication rules
(∗) above it is impossible to apply such a rule to go from a monomial like m0 ·Q[X0 ∪X1 ] m1
or m1 ·Q[X0 ∪X1 ] m0 with an occurrence of a factor of type (i)-(ix) above to a monomial m
without a factor of type (i)-(ix). We leave this tedious but easy verification to the reader.
Our claim now follows by ∆01 -induction on the number of applications of (∗). This completes
the proof that I1 and Q1 are computable. Furthermore, it follows from our previous results
in constructing Q1 that every nonzero monomial m ∈ Q1 either lies in Q0 ,→ Q1 or else it
22
CHRIS J. CONIDIS
has a unique monomial representative m ∈ Q[X0 ∪ X1 ] of the form
k
k
,
Ak , YAA01 YAA12 · · · YAAk−1
YAA01 YAA12 · · · YAAk−1
or
k
k
· · · ZAA12 ZAA01 ,
· · · ZAA12 ZAA01 , ZAAk−1
Ak ZAAk−1
for some k ∈ N and A0 , . . . , Ak ∈ M0 such that ind(A0 ) < ind(A1 ) < · · · < ind(Ak ).
Moreover every monomial of one of these forms corresponds to a nonzero monomial in Q1 .
It now follows that hX0 iQ1 and hX0 ∪ X1 iQ1 are computable ideals of Q1 such that hX0 iQ1 ⊂
hX0 ∪X1 iQ1 . Furthermore, it is easy to check that if m0 , m1 ∈ Q[X0 ∪X1 ] are unique monomial
representatives as displayed above, then m0 is a left/right divisor of m1 in Q1 if and only if
m0 is a left/right divisor of m1 in Q[X0 ∪ X1 ]. It follows that the set of left/right divisors
of m1 in Q1 is computable, uniformly in m1 and the left/right condition. In other words,
for all unique monomial representatives m ∈ Q[X0 ∪ X1 ], the sets of left and right monomial
multiples of m in Q1 are computable, uniformly in m and the left/right condition.
Now, we claim that xnx +1 =Q1 0 for all x ∈ hX0 ∪ X1 iQ1 , where nx ∈ N is the number
of summands of x when x is written as a unique linear combination of the nonzero unique
monomial representatives in Q1 displayed in the previous paragraph. Our claim follows from
the finitary pigeonhole principle and the subclaim that if m0 , m1 , m2 , . . . , mk , k ∈ N, are such
unique monomial representatives and mi = mj for some 0 ≤ i < j ≤ k, then the product
m0 m1 m2 · · · mk equals zero in Q1 . The subclaim follows easily from our characterization
of the unique monomial representatives of Q1 displayed in the previous paragraph. For all
YBA , ZBA ∈ X1 , A, B ∈ M0 , ind(A) > ind(B), let ind(YBA ) = ind(ZBA ) = ind(A) ∪ ind(B) ⊂ N.
Similarly, for all monomials m ∈ Q[X0 ∪ X1 ] define ind(m) ⊂ N to be the union of ind(x),
where x ranges over all indeterminate factors of m. This completes our construction of Q1 .
We now turn our attention to constructing every Qm+1 ⊃ Qm , m ∈ N, m > 0.
Constructing Qm+1 ⊃ Qm for all m ∈ N, m > 0:
Preliminaries and the structure of Qm :
We now describe the structure of Qm , m ∈ N, before we give the construction of Qm+1 ⊃
Qm .
Suppose that m ∈ N, m > 0, is given and that the computable ring Qm has been constructed as a quotient ring of the noncommutative computable polynomial ring
Q[X0 ∪ X1 ∪ · · · ∪ Xm ]
such that
Q0 ,→ Q1 ,→ · · · ,→ Qm ,
where Qk ,→ Qk+1 , 0 ≤ k < m, is induced via the natural embedding
Q[X0 ∪ X1 ∪ · · · ∪ Xk ] ,→ Q[X0 ∪ X1 ∪ · · · ∪ Xk+1 ], 0 ≤ k < m,
and
hX0 iQm ⊂ hX0 ∪ X1 iQm ⊂ · · · ⊂ hX0 ∪ X1 ∪ · · · ∪ Xm iQm ,
where the ideals hX0 ∪ X1 ∪ · · · ∪ Xk iQm are (uniformly) computable for all 0 ≤ k ≤ m,
and the sets of indeterminates X0 , X1 , . . . , Xm are disjoint. Suppose also that every x ∈
hX0 ∪ X1 ∪ · · · ∪ Xm iQm is nilpotent in Qm .
Assume also that ind(M ) ⊂ N has been defined for every monomial M ∈ Q[X0 ∪ X1 ∪
· · · ∪ Xm ]. Also suppose that X0 , X1 , . . . , Xm are disjoint sets of indeterminates such that for
all 0 < k ≤ m every indeterminate X ∈ Xk is of the form
• X = YBA , or
• X = ZBA
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
23
for some type of monomial
A ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xk−1 ] \ Q[X0 ∪ X1 ∪ · · · ∪ Xk−2 ]
to be constructed in the next paragraph (here X−1 = ∅), and indeterminate
B ∈ X0 ∪ X1 ∪ · · · ∪ Xk−1
such that either
• ind(A) > ind(B), or else
• B ∈ Xj , 0 ≤ j < m, and A ∈
/ hX0 ∪ X1 ∪ · · · ∪ Xj iQk−1 (or, equivalently, A ∈
/
hX0 ∪ X1 ∪ · · · ∪ Xj iQ[X0 ∪X1 ∪···∪Xm ] ).
Now, assume that every nonzero monomial m0 ∈ Qm , m0 ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xm ], has
a unique nonzero monomial representative of the following form. Via Σ01 -induction,
let 0 ≤ k < m be the least number such that m0 ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xk+1 ] \ Q[X0 ∪
X1 ∪ · · · ∪ Xk ] (this is equivalent to saying that 0 ≤ k ≤ m is the least number such that
m0 ∈ (Qk+1 \ Qk ) ,→ Qm ). Assume that we have already specified unique nonzero monomial
representatives m0 ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xk ] for all monomials m0 ∈ Qk and also that the sets
of left and right monomial multiples in Qk for all such m0 ∈ Qk are computable, uniformly
in m0 and the left/right condition. Then m0 ∈ Qm has a unique monomial representative
m0 ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xm ] that divides a monomial m̂0 ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xk+1 ] of the
form:
A
A
A
A
~ A1 q 1 Z
~ A3 q2 · · · qj−1 Z
~ 2j−1 qj Y~ 2j+1 qj+1 Y~ 2j+3 qj+2 · · · ql−1 Y~ 2l−1 ql
m̂0 = q0 Z
A2j+2
A2l−2
A2j
A0
A2
A2j−2
(?)
such that:
• j, l ∈ N, j < l, l > 1;
• Each q0 , q1 , . . . , ql ∈ Q[X0 ∪X1 ∪· · ·∪Xk ] is a unique nonzero monomial representative
(that we specified above) for corresponding q 0 , q 1 , . . . , q l ∈ Qk , or we may also have
that q0 = 1 ∈ Q or ql = 1 ∈ Q;
• Y~BA = YBC0 YCC01 · · · YCAn , B, Ci ∈ Xk indeterminates, 0 ≤ i ≤ n, and A ∈ Q[X0 ∪ X1 ∪
· · · ∪ Xk ] a unique nonzero monomial representative for Qk such that:
– ind(C0 ) < ind(C1 ) < · · · < ind(Cn ) < ind(A);
– If ind(B) ind(C0 ) then B ∈ X0 ∪ X1 ∪ · · · ∪ Xi for some 0 ≤ i < k such that
C0 ∈
/ hX0 ∪ X1 ∪ · · · ∪ Xi iQk (or, equivalently, C 0 ∈
/ hX0 ∪ X1 ∪ · · · ∪ Xi iQm );
C
C
A
A
0
n
~ = Z Z · · · Z , B, Ci ∈ Xk , 0 ≤ i ≤ n, and A ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xk ] a unique
• Z
B
C0 C1
B
nonzero monomial representative for Qk such that:
– ind(A) > ind(C0 ) > ind(C1 ) > · · · > ind(Cn );
– If ind(B) ind(Cn ) then B ∈ X0 ∪ X1 ∪ · · · ∪ Xi for some 0 ≤ i < k such that
Cn ∈
/ hX0 ∪ X1 ∪ · · · ∪ Xi iQk (or, equivalently, C n ∈
/ hX0 ∪ X1 ∪ · · · ∪ Xi iQm );
• A1 is a left monomial multiple of q 0 in Qk ;
• For all 0 < i < j we have that A2i+1 is a left monomial multiple of A2i−2 q i in Qk ;
• A2j−2 qj A2j 6=Qk 0;
• For all j < i < l we have that A2i−1 is a right monomial multiple of q i A2i in Qk ;
• A2l−1 is a right monomial multiple of ql in Qk ;
For all k = 0, 1, 2, . . . , m let Mk ⊆ Q[X0 ∪ X1 ∪ · · · ∪ Xm ] denote the set of nonzero monomials
of (Qk \ Qk−1 ) ,→ Qm (here Q−1 = ∅) described above. Also assume that every x ∈
hX0 ∪ X1 ∪ · · · ∪ Xm iQm is nilpotent in Qm .
We have now specified all of the information that we require about Qm and its subrings
and ideals. Note that Q1 ⊃ Q0 satisfies all of these hypotheses. Later on we will use these
remarks (about Qm ) and our constructions of Q1 ⊃ Q0 and Qm+1 ⊃ Qm , m ∈ N, along with
mathematical induction to extend our remarks about Qm to the ring Q that we will define
24
CHRIS J. CONIDIS
via
Q=
[
Qm .
m∈N
We are now ready to give the construction of Qm+1 ⊃ Qm .
The construction of Qm+1 :
We begin by setting
Y0m+1 = {YBA : A, B ∈ Mm , ind(A) > ind(B)},
Z0m+1 = {ZBA : A, B ∈ Mm , ind(A) > ind(B)},
/ hX0 , X1 , . . . , Xk iQ[X0 ∪X1 ∪···∪Xm ] },
Y1m+1 = {YBA : A ∈ Mm , B ∈ Mk , k < m, A ∈
Z1m+1 = {ZBA : A ∈ Mm , B ∈ Mk , k < m, A ∈
/ hX0 , X1 , . . . , Xk iQ[X0 ∪X1 ∪···∪Xm ] },
Ym+1 = Y0m+1 ∪ Y1m+1 ,
Zm+1 = Z0m+1 ∪ Z1m+1 ,
and finally
Xm+1 = Ym+1 ∪ Zm+1 .
Xm+1 will be the new indeterminates of Qm+1 . Let Qm [Xm+1 ] be the computable noncommutative polynomial ring with coefficients in Qm and indeterminates in Xm+1 . Then Qm+1
will be a quotient ring of the form
Qm+1 = Qm [Xm+1 ]/I,
for some computable ideal I ⊂ Qm [Xm+1 ]. We say that m ∈ Qm [Xm+1 ] is a monomial
whenever m is a monomial in Q[Xm ], Qm or m is a product of monomials from Qm and
Q[Xm+1 ].
0
⊆ Qm [Xm+1 ] to be
As in the construction of Q1 ⊃ Q0 above, we define the ideal Im+1
generated by the set
{YBA AZBA − B : A, B ∈ Mm , YBA ∈ Ym+1 , ZBA ∈ Zm+1 as above}.
As before these generators correspond to multiplication rules of the form:
(∗)
YBA AZBA = B, A, B ∈ Mm , YBA ∈ Ym+1 , ZBA ∈ Zm+1 as above,
0
in the quotient ring Qm [Xm+1 ]/hIm+1
iQm [Xm+1 ] . Furthermore, as before we say that a monomial m ∈ Qm [Xm+1 ] is reduced whenever m contains no factors of the form YBA AZBA , where
A, B ∈ Mk , 0 ≤ k ≤ m, YBA ∈ Ym+1 ,ZBA ∈ Zm+1 as above. By our choice of representative for monomials in Qm (?) above it follows that every factor of every monomial
m ∈ Qm ,→ Qm [Xm+1 ] is already reduced. As before when we constructed Q1 (i.e. via a similar proof it follows that) we have that every monomial m ∈ Qm [Xm+1 ] has a unique reduced
monomial representative m̂ ∈ Qm [Xm+1 ] and that m̂ is obtained effectively from m by first
substituting B for all factors of the form YBA AZBA , YBA ∈ Ym+1 ,ZBA ∈ Zm+1 , A, B ∈ Mm , in
m (which do not overlap) to obtain a monomial m0 ∈ Qm [Xm+1 ], and then expressing every
resulting monomial factor of m0 that lives in Qm in terms of its unique nonzero monomial
representative (?) defined above. The important details were given during the construction
of Q1 above, and so we leave this verification to the reader. More precisely, by our previ0
ous remarks it follows that every monomial m ∈ Qm [Xm+1 ]/hIm+1
iQm has a unique reduced
monomial representative m ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xm+1 ] of the form
m = q0 m0 q1 m1 q2 · · · qk mk ,
where k ∈ N, qi ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xm ] is a unique nonzero monomial representative inside
Qm , for all 0 ≤ i ≤ k, and mi ∈ Q[Xm+1 ] is a reduced monomial for all 0 ≤ i ≤ k, such that
no factor of the form YBA AZBA , YBA ∈ Ym+1 ,ZBA ∈ Zm+1 , A, B ∈ Mm , appears in m. We also
include the possibility that either q0 or mk are equal to 1 ∈ Q ,→ Q[X0 ∪ X1 ∪ · · · ∪ Xm+1 ].
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
25
As in the construction of Q1 above, let
0
Q0m+1 = Qm [Xm+1 ]/hIm+1
iQm [Xm+1 ]
and let red(Q0m+1 ) ⊂ Q[X0 ∪ X1 ∪ · · · ∪ Xm+1 ] denote the set of reduced monomial representatives constructed in the previous paragraph. By our remarks in the previous paragraph
it follows that Q0m+1 is a computable ring and red(Q0m+1 ) is a computable set of monomials. Now, because every indeterminate YBA ∈ Ym+1 ,ZBA ∈ Zm+1 , appears in exactly one
0
0
has an
⊂ Qm [Xm+1 ], and also every generator in Im+1
element of the generating set Im+1
A
A
appearance of some YB ∈ Ym+1 and also some ZB ∈ Zm+1 , it follows that Qm ,→ Q0m+1
0
(i.e. hIm+1
iQm [Xm+1 ] contains no elements in Qm ). Next we construct an ideal Im+1 ⊂ Q0m+1
such that Qm+1 = Q0m+1 /Im+1 . This will complete our construction of Qm+1 and essentially
complete our construction of Q = Q∞ = ∪k∈N Qk .
For all monomials a, b ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xm ] let ldm (a, b) denote the fact that a is
a left divisor of b in Qm , and let rdm (a, b) denote the fact that a is a right divisor of
b in Qm . Let ¬ldm (a, b) and ¬rdm (a, b) denote the negations of ldm (a, b) and rdm (a, b),
respectively. Recall that (by our previous remarks) the relations ldm (a, b),rdm (a, b) and
their negations are uniformly computable in the monomials a, b ∈ Q[X0 ∪ X1 ∪ · · · Xm ].
Finally, let Im+1 be the ideal of Q0m+1 generated by the images m ∈ Q0m+1 of monomials
m ∈ red(Q[X0 ∪ X1 ∪ · · · ∪ Xm+1 ]) such that m contains a factor of the form:
(i) Y Z, Y ∈ Ym+1 ⊂ Xm+1 , Z ∈ Zm+1 ⊂ Xm+1 ; or
(ii) YBA X, A, B, X ∈ Mm , YBA ∈ Ym+1 , ¬rdm (X, A); or
(iii) YBA XYDC , A, B, C, D, X ∈ Mm , YBA , YDC ∈ Ym+1 , ¬rdm (XD, A); or
(iv) XZBA , A, B, X ∈ Mm , ZBA ∈ Zm+1 , ¬ldm (X, A); or
C
C
(v) ZD
XZBA , A, B, C, D, X ∈ Mm , ZBA , ZD
∈ Zm+1 , ¬ldm (DX, A); or
A
C
A
(vi) YB XZD , A, B, C, D, X ∈ Mm , YB ∈ Ym+1 , ZBA ∈ Zm+1 , such that either A 6= C or
B 6= D or X 6= A; or
(vii) YBA YDC , A, B, C, D ∈ Mm , YBA , YDC ∈ Ym+1 , A 6= D; or else
C
C
∈ Zm+1 , B 6= C.
, A, B, C, D ∈ Mm , ZBA , ZD
(viii) ZBA ZD
This completes our construction of Qm+1 . We now show that Qm+1 is a computable ring
- i.e. we prove that the ideal Im+1 ⊆ Q0m+1 generated by the monomials with factors given
in (i)-(viii) above is in fact computable. Note that Qm ,→ Qm+1 since Qm ,→ Q0m+1 and
every monomial generator (i)-(viii) above contains a factor in Xm+1 . Also note that Qm+1
is a computable ring, since Im+1 ⊂ Q0m+1 is a computable ideal, which follows from the
elementary fact that if m ∈ red(Q[X0 ∪ X1 ∪ · · · ∪ Xm+1 ]) is a monomial that is divisible by
a factor of type (i)-(viii) above, then so is any reduced (left/right) monomial multiple of m
(we leave the verification of this elementary fact to the reader).
We now claim that
hX0 iQm+1 ⊂ hX0 ∪ X1 iQm+1 ⊂ · · · ⊂ hX0 ∪ X1 ∪ · · · ∪ Xm+1 iQm+1 .
Our claim follows from our subclaim that every nonzero monomial m ∈ Qm+1 has a unique
nonzero monomial representative m ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xm+1 ] of the form (?) above with
m + 1 ∈ N replacing m ∈ N via our multiplication rules (∗) above and our construction of
Im+1 also above. It follows from our subclaim that A ∈
/ hX0 ∪ X1 ∪ · · · ∪ Xi ]iQm+1 for all
(monomial) indeterminates A ∈ Xi+1 , 0 ≤ i < m + 1. It is not difficult to verify our subclaim
(indeed we essentially constructed the ideal Im+1 ⊂ Q0m+1 , Qm+1 = Q0m+1 /Im+1 , so that our
subclaim holds) and so we leave its verification to the reader.
Now, using the hypothesis that every element x ∈ hX0 ∪ X1 ∪ · · · ∪ Xm iQm is nilpotent we
will show that every element x ∈ hX0 ∪ X1 ∪ · · · ∪ Xm+1 iQm+1 is nilpotent. More precisely we
now assume that every monomial m0 ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xm ] that contains an indeterminate
divisor X 0 ∈ X0 ∪ X1 ∪ · · · ∪ Xm of multiplicity 2 is equal to zero in Qm (recall that, by
our previous remarks, this is the case when m = 1) and we now show that every monomial
26
CHRIS J. CONIDIS
m ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xm+1 ] that contains an indeterminate divisor X ∈ X0 ∪ X1 ∪ · · · ∪ Xm+1
of multiplicity 2 equals zero in Qm+1 . Let m ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xm+1 ] be a monomial that
contains an indeterminate divisor X ∈ X0 ∪ X1 ∪ · · · ∪ Xm+1 of multiplicity 2, and by our
assumption above we can assume that X = YBA or X = ZBA for some YBA ∈ Ym+1 , ZBA ∈ Zm+1 ,
A, B ∈ Mm . Without loss of generality assume that X = YBA (the case X = ZBA is similar).
Furthermore, by our characterization of the unique nonzero monomial representatives in
Qm+1 above, we can assume that a factor of the form
m1 = YBA r0 a1 r2 a3 · · · r2k a2k+1 YBA ∈ Q[X0 ∪ X1 ∪ · · · ∪ Xm+1 ]
divides m where ri is a nonzero monomial in Q[X0 ∪ X1 ∪ · · · ∪ Xm ], 0 ≤ i ≤ 2k even, and ai
is a nonzero monomial in Q[Xm+1 ] of the form
A
A
Al
,i
1,i
2,i
ai = YA0,i
YA1,i
· · · YAl i−1,i , li ∈ N, 0 ≤ i ≤ 2k + 1 odd,
i
except for a2k+1 which may equal 1 ∈ Q. Then, by our multiplication rules (∗) above and
the way in which we constructed the (indices of the) subscripts/superscripts in
Ym+1 = Y0m+1 ∪ Y1m+1
above,
(1)
(2)
(3)
it follows that we cannot have that
rd(r0 , YBA ); and
rd(ri , Ali−1 ,i−1 ) for all 2 ≤ i ≤ 2k even; and
r2k ·Qm A0,2k+1 6=Qm 0 (this translates to r2k ·Qm B 6=Qm 0 in the case when a2k+1 =
1 ∈ Q).
By the previous sentence and the way in which we constructed Im+1 ⊂ Q0m+1 above, it follows
that
m =Qm+1 m1 =Qm+1 0 ∈ Qm+1 .
Now, by the same reasoning that we gave during the construction of Q1 above, i.e. by the
finitary pigeonhole principle and our remarks in the previous few sentences, it follows that
xnx +1 =Qm+1 0,
for all x ∈ hX0 ∪ X1 ∪ · · · ∪ Xm+1 iQm+1 , where nx ∈ N is the number of nonzero monomial
summands of x ∈ Qm+1 . Also, by our previous remarks in this paragraph and the way in
which we constructed our indeterminates X0 ∪ X1 ∪ · · · ∪ Xm+1 at all previous stages it follows
that
(∗∗) ∀j, k ∈ N hx ∈ X0 ∪X1 ∪· · · Xk : ind(x) ≤ jiR0 is a finite dimensional Q−vector space.
We will use this fact (∗∗) later on when we show that every infinite strictly descending chain
of ideals in R computes the Halting Set ∅0 .
Constructing Q = Q∞ = ∪m∈N Qm :
We leave it to the reader to verify that our construction of
Q0 , Q1 , Q2 , . . . , Qm , . . .
is uniformly computable in m ∈ N. It follows that the ring
[
Qm
Q = Q∞ =
m∈N
exists via RCA0 . Furthermore, Q can be thought of as a computable quotient of
Q[X0 ∪ X1 ∪ X2 ∪ · · · ∪ Xm ∪ Xm+1 ∪ · · · ] → Q, m ∈ N.
Furthermore, by mathematical induction and/or the well-ordering principle for N and our
previous remarks in the constructions of Q1 (the base case) and Qm+1 (the induction step)
above, it follows that:
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
27
(1) Qm ,→ Q, for all m ∈ N, and the injection is induced by the natural map
Q[X0 ∪ X1 ∪ · · · ∪ Xm ] ,→ Q[X0 ∪ X1 ∪ · · · ∪ Xm ∪ Xm+1 ∪ · · · ];
(2) Every nonzero monomial element of Qm ,→ Q has a unique nonzero monomial representative, as described in (?) above;
(3) hX0 ∪ X1 ∪ · · · ∪ Xm iQ ⊂ hX0 ∪ X1 ∪ · · · ∪ Xm ∪ Xm+1 iQ , for all m ∈ N;
(4) Every element of hX0 ∪ X1 ∪ · · · ∪ Xm iQ is nilpotent, for every m ∈ N.
When taken together, (3) and (4) above imply that the unique maximal ideal of Q is
M = hX0 ∪ X1 ∪ · · · ∪ Xm ∪ Xm+1 ∪ · · · iQ .
More precisely, it follows from (4) above that every element in Q \ M is invertible in Q,
since every element in Q \ M is the image of a polynomial in Q[X0 ∪ X1 ∪ X2 ∪ · · · ] with a
nontrivial constant summand, and hence in Q this element is of the form x = q + x0 ∈ Q,
where 0 6= q ∈ Q, and x0 ∈ M is nilpotent. Now, from basic algebra it follows that x ∈ Q
is invertible in Q. On the other hand, by (3) we know that hX0 ∪ X1 ∪ · · · ∪ Xm iQ 6= Q for
all m ∈ N. It follows that M is a proper ideal of Q, and by our previous remarks in this
paragraph it follows that M is a maximal ideal of Q and that M is the unique ideal of Q
with this property. Also note that M is not nilpotent since we have that
k
YXX01 YXX12 YXX23 · · · YXXk−1
6=Q 0,
for all k ∈ N, by either (1) or (2) above.
Constructing R ⊂ Q:
Now, we will construct our desired ring R mentioned in the statement of the current
proposition above as a computably enumerable subring of Q. For all i ∈ N let Γi ∈ N,
Γi = lims Γsi , be larger than the settling time of a fixed uniform computable approximation
to the first i bits of ∅0 (obtained via a movable marker construction; see [Soa87] for more
details). Furthermore, without loss of generality we can assume that we have
(i) Γi+1 ≥ 2Γi , i ∈ N;
(ii) Γ0 ≥ 1;
(iii) whenever x, y ∈ N are such that there exist i, s such that Γsi ≤ x, y < Γsi+1 we have
that at all stages t ≥ s there exists i0 ∈ N, i0 ≤ i, such that Γti0 ≤ x, y ≤ Γti0 +1 and
hence there exists i0 ∈ N such that Γi0 ≤ x, y < Γi0 +1 .
Now, with all of this in mind we define R0 to be the c.e. subring of Q generated by nonzero
monomials m ∈ Q, m ∈ Q[X0 ∪ X1 ∪ X2 ∪ · · · ∪ Xm ∪ Xm+1 ∪ · · · ] a unique nonzero monomial
representative as in (?) above such that there exists i ∈ N so that
Γi ≤ ind(m) < Γi+1 .
Finally, via Lemma 2.7 in Section 2.3 above, let R be a computable ring that is computably
isomorphic to R0 ⊂ Q. Since R and R0 are computably isomorphic. We will identify these
rings throughout the rest of this article.
It is not difficult to see (the verification is similar to that for Q given above) that R contains
a unique maximal ideal M ⊂ R corresponding to M ∩ R0 ⊂ Q. Moreover, it follows that M
is not nilpotent since M ∩ R0 is not nilpotent. To see why M ∩ R0 is not nilpotent, let n ∈ N
be given, and let in ∈ N be such that Γin +1 − Γin > n (in exists via mathematical induction
and our construction of Γi , i ∈ N, given above). Now, the indeterminate monomials
XΓ
+1
XΓ
+2
XΓi
YXΓ in , YXΓ in+1 , · · · , YXΓ
in
in
n+1
in+1
−1
−2
∈ Y01 ⊂ X1
28
CHRIS J. CONIDIS
are all in M ∩ R0 (by our construction of R0 above) and hence so is the product
XΓ
+1
XΓ
+2
XΓi
−1
m0 = Y XΓin Y XΓin +1 · · · Y XΓ n+1 −2 6=R0 0.
in
in
in+1
We know that this product exists and is nonzero in R0 since it exists and is nonzero in Q,
by our characterization of the nonzero monomials in Q (i.e. item (2) in the second to last
list above). Now, since 0 6=R0 m0 ∈ (M ∩ R0 )n , it follows that (M ∩ R0 )n 6=R0 0. Recall that
n ∈ N is an arbitrary fixed natural number. Hence, M ∩ R0 is not nilpotent, as we claimed.
It is also not difficult to verify that R contains an infinite strictly ascending uniformly
computable chain of ideals
where Ii+1
{0} = I0 ⊂ I1 ⊂ I2 ⊂ · · · ⊂ R,
⊂ R, i ∈ N, corresponds to the ideal Iˆi ⊂ R0 given by
Iˆi = hX0 ∪ X1 ∪ · · · ∪ Xi iR0 = hX0 ∪ X1 ∪ · · · ∪ Xi iQ ∩ R0 .
We also have that every monomial m ∈ R0 , m ∈ Q[X0 ∪X1 ∪· · · ] a unique nonzero monomial
representative described by (?) above, satisfies m =R0 0 whenever there does not exist i ∈ N
such that Γi ≤ ind(m) < Γi+1 . This follows from our construction of R above and the fact
that for all indeterminates X 0 , X 1 ∈ X0 ∪X1 ∪· · · such that ind(X 0 ) < ind(X 1 ) we have that
0 1
1 0
0
and Q given above).
X X =Q X X =Q 0 (which follows from our construction of Im+1
All that is left to do is verify that every infinite strictly descending chain of ideals in
R∼
= R0 ⊂ Q computes the Halting Set ∅0 . We do this now.
Verifying that every strictly descending chain of ideals in R ∼
= R0 ⊂ Q computes
the Halting Set ∅0 :
Let J0 = R ⊃ J1 ⊃ J2 ⊃ · · · be an infinite strictly descending chain of ideals in R, and let
x̂0 , x̂1 , x̂2 , . . . ∈ R be an infinite sequence of elements such that x̂i ∈ Ji \ Ji+1 for all i ∈ N.
Let x0 , x1 , x2 , . . . ∈ R0 be the sequence corresponding to x̂0 , x̂1 , x̂2 , . . . under the isomorphism
R∼
= R0 . By our construction of R0 and R above, it follows that the sequence x0 , x1 , x2 , . . .
is computable from the sequence x̂0 , x̂1 , x̂2 , . . .. Also, since M ⊂ R0 is the unique maximal
ideal of R0 , it follows that for all i ≥ 1 we have that xi ∈ R0 is not a unit and xi ∈ M.
Now, nonuniformly choose z0 ∈ M ⊂ R0 such that the corresponding element ẑ0 ∈ M ⊂ R
satisfies ẑ ∈
/ J2 , and let k0 ∈ N be such that z0 ∈ hX0 ∪ X1 ∪ X2 ∪ · · · ∪ Xk0 iR0 . Now, via
bounded ∆02 -comprehension, which is equivalent to the Σ02 bounding principle BΣ2 (which is
provable via IΣ2 ), let 0 ≤ k1 ≤ k0 be such that the ideal corresponding to
hX0 ∪ X1 ∪ · · · ∪ Xk1 iR0
in R ∼
= R0 is a subset of Ji for all i ∈ N. By our definition of k1 ∈ N it follows that we may
assume that every xi ∈ R0 , i ∈ N, corresponds to an element x̂i ∈ R such that every nonzero
monomial summand of x̂i is not a member of the ideal corresponding to hX0 ∪X1 ∪· · ·∪Xk1 iR0
in R. Let n0 ∈ N be such that for all n ≥ n0 we have that the ideal in R corresponding to
hX0 ∪ X1 ∪ · · · ∪ Xk1 iR0
is in Jn , but the ideal in R corresponding to
hX0 ∪ X1 ∪ · · · Xk1 +1 iR0
is not in Jn . From now on we will primarily work with the sequence of elements
xn0 , xn0 +1 , xn0 +2 , . . . ∈ R0 .
First of all, note that every xn0 +n ∈ R0 , n ∈ N, must have a nonzero monomial summand
mn0 +n ∈ R0 in the set
hX0 ∪ X1 ∪ X2 ∪ · · · ∪ Xk1 +1 iR0 \ hX0 ∪ X1 ∪ X2 ∪ · · · ∪ Xk1 iR0 .
A NEW PROOF THAT ARTINIAN IMPLIES NOETHERIAN
29
Otherwise, we would have that some xn0 +n , n ∈ N, has a nonzero monomial summand of
the form
mn0 +n ∈ M \ hX0 ∪ X1 ∪ · · · ∪ Xk1 +1 iR0 ,
and we would have constructed indeterminates
[
[
m
m
YB n0 +n ∈
Ym1 and ZB n0 +n ∈
Z1m
m∈N
m≥2
m∈N
m≥2
such that mn0 +n ∈ Q[X0 ∪ X1 ∪ · · · ] is the unique nonzero monomial representative of mn0 +n
in R0 described by (?) above, and B is any indeterminate/element in/of X0 ∪X1 ∪Xk1 ∪Xk1 +1 .
It follows that the ideal in R ∼
= R0 corresponding to
hX0 ∪ X1 ∪ · · · ∪ Xk1 +1 iR0
is included in Jn0 +n , contradicting our definition of n0 above. Therefore, every xn0 +n , n ∈ N,
must have a nonzero monomial summand mn0 +n ∈ R0 in the set
hX0 ∪ X1 ∪ X2 ∪ · · · ∪ Xk1 +1 iR0 \ hX0 ∪ X1 ∪ X2 ∪ · · · ∪ Xk1 iR0 .
Now, let
mn0 , mn0 +1 , mn0 +2 , . . . ∈ hX0 ∪ X1 ∪ X2 ∪ · · · ∪ Xk1 +1 iR0 \ hX0 ∪ X1 ∪ X2 ∪ · · · ∪ Xk1 iR0
be the infinite sequence of nonzero monomials constructed in the previous paragraph, and
for all n ∈ N let mn0 +n ∈ Q[X0 ∪ X1 ∪ · · · ] be the unique nonzero monomial representative of
mn0 +n described by (?) above. Let n0 ≤ i0 < i1 < i2 < · · · be an infinite strictly increasing
sequence of numbers such that
ind(mi0 ) < ind(mi1 ) < ind(mi2 ) < · · · .
The fact that the infinite sequence i0 < i1 < i2 < · · · exists follows from our remark (∗∗)
above and the fact that every ideal Jk , k ∈ N, corresponds to a Q-subspace of R. Now, by
our construction of indeterminates in X0 ∪ X1 ∪ X2 ∪ · · · , our multiplication rules (∗) above,
and the fact that xn0 +n ∈ Jn0 +n \ Jn0 +n+1 , for all n ∈ N, it follows that for all k ∈ N there
exists Γjk ∈ N such that
ind(mik ) < Γjk ≤ ind(mik+1 ).
It follows that the uniformly computable (relative to ⊕∞
i=0 Ji ) sequence of numbers
c0 = min(ind(mi0 )) < c1 = min(ind(mi1 )) < c2 = min(ind(mi2 )) < · · ·
satisfies cj ≥ Γj for all j ∈ N. Hence, from the sequence {cj }j∈N (that is uniformly computable in {Jk }k∈N ) we can compute the Halting Set ∅0 .
Finally, note that by our construction of R it follows that R contains an infinite ∅0 computable strictly descending chain of ideals
J0 ⊃ J1 ⊃ J2 ⊃ · · · ,
where Ji , i ∈ N, is generated by the set of monomials m ∈ R that satisfy
ind(m) ⊆ {Γi , Γi + 1, Γi + 2, . . .} ⊆ N.
This completes our construction of R, and marks the end of the proof of Theorem 5.3. 30
CHRIS J. CONIDIS
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Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1,
CANADA
E-mail address: [email protected]