WHEN A SUPER-DECOMPOSABLE PURE-INJECTIVE
MODULE OVER A SERIAL RING EXISTS
GENA PUNINSKI
Abstract. We give a criterion for the existence of a super-decomposable
pure-injective module over an arbitrary serial ring.
1. Introduction
A module M is said to be super-decomposable if M does not contain
indecomposable direct summands.
The existence of a super-decomposable pure-injective module over a given
ring R is a natural question in the theory of modules. The answer depends
on a variety of ranks defined on the lattice of all pp-formulae over R and
may be considered as a measure of complexity of the entire category of
R-modules. For instance, according to Trlifaj [10], over a von Neumann
regular ring R, a super-decomposable (pure) injective module exists iff R
is not semiartinian. Rather few cases are known where such a satisfactory
answer can be obtained. For example Puninski (see [7, Ch. 12]) proved that
over a commutative valuation ring V , a super-decomposable pure-injective
module exists iff the Krull dimension of V is undefined.
A further example we can mention is the first Weyl algebra A1 (F ) over
a field F of characteristic zero, where the existence of such a module was
proved by Prest and Puninski [6] in case F is countable. Also, as follows
from [8], every non-domestic string algebra over a countable field possesses
a super-decomposable pure-injective module.
Let us describe the situation over a serial ring R. As we have just mentioned, a complete answer is known if R is commutative. Reynders [9]
proved that a serial ring R with Krull dimension does not have a superdecomposable pure-injective module. Nevertheless, if R is a serial ring whose
2000 Mathematics Subject Classification. 03C60, 16D50, 16L30.
Key words and phrases. Serial ring, super-decomposable module, pure-injective
module.
This paper was written during the visit of the author to the University of Manchester
supported by EPSRC grant GR/R44942/01. He would like to thank the university for the
kind hospitality.
1
Krull dimension is undefined, it is not always the case that R possesses such
a module. For instance, if R is an exceptional uniserial ring, then (see [7,
Ch. 15]) R has neither Krull dimension nor a super-decomposable pureinjective module.
In this paper we give a simple criterion for when a super-decomposable
pure-injective module over a serial ring exists. In particular it will follow that
if there is a super-decomposable pure-injective right module over a serial ring
R, then there is a super-decomposable pure-injective left R-module. It is
known that, over an arbitrary ring R, if a super-decomposable pure-injective
R-module exists, then the lattice of all pp-formulae over R does not have
width. We prove that for serial rings the converse is also true. Using this
criterion we also show that a serial ring whose lattice of two-sided ideals has
Krull dimension fails to have a super-decomposable pure-injective module
— a result that seems to be nearly the best possible.
2. Preliminaries
The notions from ring and module theory used in this paper are quite
standard. Recall that a module M is said to be uniserial if the lattice of
submodules of M is a chain, and M is serial if M is a direct sum of uniserial
modules. As usual, a ring R is serial if both RR and R R are serial modules.
This just means that the unit of R can be decomposed into orthogonal
pieces 1 = e1 + · · · + en such that every (right) module ei R is uniserial and
every (left) module Rei is uniserial. For instance, the ring Tn (F ) of upper
triangular matrices over a field F is serial (and artinian). Note that every
serial ring is semiperfect.
Let us recall some basic notions of the model theory of modules.
Let R be a ring with a unit, and let LR be a language of right R-modules.
A positive-primitive (pp-) formula φ(x) is an existential LR -formula ∃ ȳ =
(y1 , . . . , yk ) (ȳA = xb̄), where A is a k × l matrix over R, and b̄ is a row
of length l with entries in R. We use A | xb̄ (A divides xb̄) as a shorthand
for this formula. For instance, a | x is an abbreviation for the pp-formula
∃ y (ya = x) which expresses that x is divisible by a.
Let M be a right R-module, and let φ(x) be a pp-formula. We write
M |= φ(m) if there exists a tuple n̄ = (n1 , . . . , nk ) ∈ M such that n̄A = mb̄.
Then φ(M ) = {m ∈ M | M |= φ(m)} is a left S-submodule of M , where
S = End(M ). For instance, (a | x)(M ) = M a, and (xb = 0)(M ) = {m ∈
M | mb = 0} is a left annihilator of b in M .
2
Given pp-formulae φ and ψ, let us define φ → ψ (φ implies ψ) if φ(M ) ⊆
ψ(M ) for every module M . For instance, ba | x → a | x for all a, b ∈ R. Also,
φ is equivalent to ψ if φ implies ψ and ψ implies φ. After factorization by
this equivalence relation the set of all pp-formulae becomes a partial order.
In fact, this poset is a modular lattice, where the meet is conjunction (of
pp-formulae), and the join is sum (of pp-formulae). Here φ + ψ is defined
as the pp-formula ∃ y φ(x − y) ∧ ψ(y). For instance, it is easily seen that the
pp-formula a | x + xb = 0 is equivalent to the pp-formula ab | xb.
There is a duality (i.e. an anti-isomorphism) D between the lattices of
right and left pp-formulae over R. In particular if φ and ψ are pp-formulae,
then φ → ψ iff Dψ → Dφ. What will be essential for us is that this duality
swaps divisibility and annihilator formulae. More precisely, D(a | x) is the
left annihilator formula ax = 0, and D(xb = 0) is the left divisibility formula
b | x.
A free realization of a pp-formula φ is a pair (M, m) where M is a finitely
presented module, m ∈ M is such that 1) M |= φ(m) and 2) if M |= ψ(m)
for some pp-formula ψ, then φ → ψ. Every pp-formula has a free realization.
For instance, (R/aR, b) is a free realization of the formula ∃ y (ya = 0 ∧ yb =
x).
A pp-type is a set of pp-formulae closed with respect to finite conjunctions
and implications. If m is an element of a module M , then ppM (m) = {φ | φ
a pp-formula and M |= φ(m)} is a pp-type. Given a pp-type p, we will often
distinguish its positive part p+ from its negative part p− , consisting of all
pp-formulae which are not in p.
Additional information about the theory of serial rings and the model
theory of modules over them can be found in [7]. To make the paper uniform
we refer to this book where it is possible. Another reference for model theory
of modules is [5] and for model theory over a serial ring the paper of Eklof
and Herzog [1].
We recall some facts that will be used further without special reference.
Let e be an indecomposable idempotent of a serial ring R (hence both modules eR and Re are uniserial) and let M be either a pure-injective or a
pure-projective module over R. Then, by [7, L. 11.4], M e is distributive as
a left module over S, where S = End(M ). Moreover if M is pure-injective
and indecomposable, then M e is uniserial over S. In particular, for arbitrary pp-formulae φ(x) and ψ(x) such that φ, ψ → e | x (i.e. φ and ψ are
e-formulae), the S-submodules φ(M ) and ψ(M ) are comparable.
3
Given a, c ∈ Re, set a ≤l c if c ∈ Ra. Clearly ≤l is a linear (quasi) order.
In particular a ≤l c ≤l a holds iff Ra = Rc, and we will write a ∼l b in this
case. Therefore a <l c means that Rc ⊂ Ra. Similarly, for b, d ∈ eR, we
define b ≤r d if d ∈ bR and b <r d if dR ⊂ bR. Also put b ∼r d if bR = dR.
According to [7, L. 11.4], the lattice Le of all e-formulae over R is generated by two chains: a | x, a ∈ Re and xb = 0, b ∈ eR. It follows from general
lattice theory (see [3, Thm. 13]) that Le is distributive, and every formula
φ ∈ Le can be represented in the form φ = (an | x + xbn = 0) ∧ . . . ∧ (a1 |
x + xb1 = 0), where ai ∈ Re, a1 <l · · · <l an and bi ∈ eR, b1 <r · · · <r bn .
Also one can rearrange brackets in the representation of φ: it is equivalent
to an | x + (xbn = 0 ∧ an−1 | x) + · · · + (xb2 = 0 ∧ a1 | x) + xb1 = 0.
If a is an element of a ring R, by RaR we will denote the two-sided ideal
generated by a. Given a, c ∈ Re, b, d ∈ eR, we write ab ≤ cd if cd ∈ RabR
and ab < cd if RcdR ⊂ RabR. Note that in general ≤ is not a linear order,
although if a ≤l c and b ≤r d (the main case we will deal with), then ab and
cd are clearly comparable with respect to ≤. Also, ab ∼ cd is an abbreviation
for RabR = RcdR.
3. pp-formulae
The following lemma describes important implications between pp-formulae
over a serial ring (to be more precise we describe a basis of open sets of the
Ziegler spectrum over R — see [9]).
Lemma 3.1. Let e be an indecomposable idempotent of a serial ring R,
a, c ∈ Re, b, d ∈ eR. Then the following are equivalent:
1) a | x ∧ xb = 0 → c | x + xd = 0;
2) a ∈ Rc or d ∈ bR or cb ≤ ad.
Proof. 2) ⇒ 1). If a ∈ Rc, then a | x → c | x, therefore the required
implication holds true. Similarly, if d ∈ bR then xb = 0 → xd = 0, and we
obtain the required conclusion.
Finally, suppose that cb ≤ ad, i.e. ad ∈ RcbR. By [5, Cor. 4.35], it suffices
to prove that in every indecomposable pure-injective module M over R we
have an inclusion
M a ∩ (xb = 0)(M ) ⊆ M c + (xd = 0)(M ).
If M cb = 0, then cb ≤ ad yields M ad = 0, hence M a ⊆ (xd = 0)(M ), which
suffices. Otherwise M cb ̸= 0 therefore M c is not contained in (xb = 0)(M ).
4
Since M e is a uniserial S-module, (xb = 0)(M ) ∩ M e ⊂ M c, which is as
desired.
Note also that if a ∈
/ Rc, d ∈
/ bR, then c ∈ Ra, b ∈ dR therefore ad ≤ cb.
Together with cb ≤ ad this yields ad ∼ cb.
1) ⇒ 2). Since Re is uniserial, there is an indecomposable idempotent f ∈
R such that Ra = Rf a. Then the formulae a | x and f a | x are equivalent.
Similarly bR = bgR for an indecomposable idempotent g ∈ R. Then the
formulae xb = 0 and xbg = 0 are equivalent. Also RabR = Rf abgR.
Thus we may assume that a ∈ f Re, c ∈ f ′ Re, b ∈ eRg and d ∈ eRg ′
for indecomposable idempotents f, f ′ , g, g ′ ∈ R. We can also assume that
a ∈
/ Rc, d ∈
/ bR. Therefore c = c′ a, c′ ∈ f ′ Rf ∩ Jac(R), and b = db′ ,
b′ ∈ g ′ Rg ∩ Jac(R).
Clearly (f R/abR, a) is a (free) realization of a | x ∧ xb = 0. This formula
implies c | x + xd = 0, which can be written as cd | xd. Then ad = hcd in
f R/abR for some h ∈ f R, i.e.
ad = abv + hcd
(1)
for some h ∈ f Rf ′ , v ∈ gRg ′ . Let us write (1) as ad = abv + hc′ ad, that is
(1 − hc′ )ad = abv. Since 1 − hc′ is invertible, it follows that ad ≥ ab. But
ab = adb′ ≥ ad, therefore ab ∼ ad.
Multiplying (1) by b′ on the right we obtain ab = abvb′ + hcb, that is
ab(1 − vb′ ) = hcb. Since 1 − vb′ is invertible, it follows that ab ≥ cb. Since
cb = c′ ab ≥ ab, we have ab ∼ cb.
Thus cb ∼ ab ∼ ad yields cb ∼ ad.
The following claims are obtained from Lemma 3.1 by substitution.
Corollary 3.2. Let e be an indecomposable idempotent of a serial ring R,
a, c ∈ Re, b ∈ eR. Then the following are equivalent:
1) a | x ∧ xb = 0 → c | x;
2) a ∈ Rc or bR = eR or cb ≤ a.
Proof. Since a | x ∧ xb = 0 is an e-formula, the implication 1) in Lemma 3.1
can be written as a | x ∧ xb = 0 → (c | x + xd = 0) ∧ e | x, and the latter is
c | x + (e | x ∧ xd = 0).
Take d = e in Lemma 3.1. Then e | x ∧ xd = 0 is equivalent to x = 0.
Therefore c | x + (e | x ∧ xd = 0) is equivalent to c | x. Finally d = e ∈ bR
in 2) means that bR = eR.
5
Corollary 3.3. Let e be an indecomposable idempotent of a serial ring R,
a ∈ Re, b, d ∈ eR. Then the following are equivalent:
1) a | x ∧ xb = 0 → xd = 0;
2) d ∈ bR or ad = 0.
Proof. Take c = 0 in Lemma 3.1. Then c | x is equivalent to x = 0, hence
c | x + xd = 0 is equivalent to xd = 0.
Now a ∈ Rc means that a = 0 (therefore ad = 0). Also cb ≤ ad is the
same as 0 ≤ ad i.e. ad = 0.
Corollary 3.4. Let e be an indecomposable idempotent of a serial ring R,
a, c ∈ Re, d ∈ eR. Then the following are equivalent:
1) a | x → c | x + xd = 0;
2) a ∈ Rc or ad = 0.
Proof. Take b = 0 in Lemma 3.1. Then xb = 0 is equivalent to x = x, hence
a | x ∧ xb = 0 is equivalent to a | x.
Now d ∈ bR means d = 0 (therefore ad = 0). Also 0 = cb ≤ ad implies
ad = 0.
Corollary 3.5. Let e be an indecomposable idempotent of a serial ring R,
c ∈ Re, b, d ∈ eR. Then the following are equivalent:
1) xb = 0 → c | x + xd = 0;
2) Rc = Re or d ∈ bR or cb ≤ d.
Proof. Note that 1) is equivalent to e | x∧xb = 0 → c | x+xd = 0. Now take
a = e in Lemma 3.1. Then a ∈ Rc means that e ∈ Rc, i.e. Re = Rc.
Now we are in a position to analyze any implication φ → ψ between e∑
formulae. Indeed we can write φ as a sum of conjuncts i ai | x ∧ xbi = 0
and write ψ as a conjunction of sums ∧j (cj | x + xdj = 0). Clearly, φ → ψ
iff ai | x ∧ xbi = 0 → cj | x + xdj = 0 holds for all i, j. Now we may apply
Lemma 3.1 to every such implication.
Let us give an example of such an analysis.
Lemma 3.6. Let e be an indecomposable idempotent of a serial ring R,
a, c ∈ Re, b, d ∈ eR. Then the following are equivalent:
1) a | x + xb = 0 → c | x + xd = 0;
2) a ∈ Rc or ad = 0, and Rc = Re or d ∈ bR or cb ≤ d.
Proof. Clearly, 1) is equivalent to a | x → c | x + xd = 0 and xb = 0 → c |
x + xd = 0. By Corollary 3.4, the first implication is the same as ‘a ∈ Rc or
ad = 0’. To conclude, apply Corollary 3.5 to the latter implication.
6
4. pp-types
In this section we describe e-types over an arbitrary serial ring. The
main point is to analyze implications of the form ∧ni=1 ai bi | xbi → ab | xb.
For a commutative valuation domain, by [7, Ch. 12], every such implication
becomes trivial: it holds iff ai bi | xbi → ab | xb holds for some i.
The following lemma is a weak form of this fact.
Lemma 4.1. Let e be an indecomposable idempotent of a serial ring R and
let a, ai ∈ Re, b, bi ∈ eR be such that ∧ni=1 ai bi | xbi → ab | xb. Then there
are i, j such that ai bi | xbi ∧ aj bj | xbj → ab | xb.
Proof. Up to equivalence of pp-formulae we may assume that a ∈ f Re,
ai ∈ fi Re, b ∈ eRg, and bi ∈ eRgi for some indecomposable idempotents
f, fi , g, gi ∈ R.
Let φi be ai | x+xbi = 0 and let φ be a | x+xb = 0. By elementary duality,
∑
→ φ yields Dφ → ni=1 Dφi . Note that Dφ is the (left) pp-formula
∧ni=1 φi
ax = 0 ∧ b | x and therefore has (M, b), M = Rg/Rab, as a free realization.
If S = End(M ) then Dφ(M ) = bS ⊆ eM . Also Mi = Dφi (M ) ⊆ eM are
∑
(right) S-submodules of M , and bS ⊆ ni=1 Mi . Since the S-module eM is
∑
distributive, it follows that bS = ni=1 (bS ∩ Mi ).
Due to [7, Prop. 2.6], S (the endomorphism ring of the uniserial module
M ) has at most two maximal ideals, hence the cyclic module bS has at
most two maximal submodules. Thus bS = (bS ∩ Mi ) + (bS ∩ Mj ) for
some i, j. Then b ∈ Mi + Mj , hence (by the definition of free realization)
Dφ → Dφi + Dφj . By duality we obtain φi ∧ φj → φ.
Now we are going to give a ‘geometric’ description of the implications of
pp-formulae in Lemma 4.1. Recall that Re is linearly ordered by a ≤l c if
c ∈ Ra. Similarly eR is linearly ordered by b ≤r d if d ∈ bR. Thus, given
a ∈ Re, c ∈ Ra, b ∈ eR, d ∈ bR (i.e. a ≤l c, b ≤r d), there are obvious
implications cb | xb → ab | xb and ab | xb → ad | xd. We will refer to these
implications as ‘trivial.’ If a formula ab | xb is drawn as a point (b, a) on the
plane eR × Re, then trivial implications act to the ‘right and down:’
7
0 O
a◦
ab | xb
◦
/
e ◦
e
/
◦
b
0
Let us describe one nontrivial implication.
Lemma 4.2. Let e be an indecomposable idempotent of a serial ring R and
let a, c ∈ Re, b, d ∈ eR be such that a ∈ Rc, b ∈ dR and ab ≤ cd. Then
ab | xb ∧ cd | xd is equivalent to ad | xd.
O
a◦
ad | xd
◦O o
ab | xb
◦
c ◦
◦
cd | xd
ab ∼ cd
◦
◦
d
◦
b
/
Proof. Using distributivity, we have (a | x + xb = 0) ∧ (c | x + xd = 0) =
(a | x ∧ c | x) + (a | x ∧ xd = 0) + (c | x ∧ xb = 0) + (xb = 0 ∧ xd = 0). Since
a ∈ Rc, the first summand is just a | x, and then the second summand can
be removed. Also b ∈ dR yields that the last summand is xd = 0.
Thus our original conjunction can be written as (a | x + xd = 0) + (c |
x ∧ xb = 0). Since ab ≤ cd, by Lemma 3.1 we obtain c | x ∧ xb = 0 → a |
x + xd = 0, which gives the desired conclusion.
An equivalence of pp-formulae as given in Lemma 4.2 will be called a
fusion of ab | xb and cd | xd.
Note that some nontrivial implications of pp-formulae can be realized as
fusions. For instance, let d <r b, a ∈ Rc and cb ≤ d. Then ab | xb → cb | xb
(trivial implication). Also e | x + xd = 0 is true everywhere, and d | xd ∧ cb |
xb → cd | xd is a fusion.
8
O
ab | xb
◦
cb ≤ d
a◦
cd | xd
◦O o
c ◦
◦
cb | xb
/
◦
◦
d
b
This remark is a part of a more general claim which completely describes
◦
the implications between e-formulae over a serial ring.
Lemma 4.3. Let e be an indecomposable idempotent of a serial ring R,
a, ai ∈ Re, b, bi ∈ eR. Then every implication ∧ni=1 ai bi | xbi → ab | xb can
be obtained as a composition of trivial implications and fusions.
Proof. If n = 1 then by Lemma 3.6 the above implication is equivalent to
a1 ∈ Ra or a1 b = 0
and Ra = Re or b ∈ b1 R or ab1 ≤ b .
If Ra = Re then ab | xb (which is the same as a | x + xb = 0) is true
everywhere. Thus we may assume that Ra ⊂ Re.
Suppose that a1 ∈ Ra. Then a1 b1 | xb1 → ab1 | xb1 is a trivial implication.
If b ∈ b1 R, then we have a trivial implication ab1 | xb1 → ab | xb. Otherwise
b <r b1 , therefore ab1 ≤ b. Then the formula e | x + xb = 0 is true
everywhere, and ab1 | xb1 ∧ b | xb → ab | xb by fusion:
0 O
a1 ◦
a◦
◦
◦O o
◦
/
◦
◦
b1
b
Thus we may assume that a1 ∈
/ Ra i.e. a1 <l a. Then a1 b = 0. If b ∈ b1 R
◦
then a1 b1 | xb1 → a1 b | xb is a trivial implication. The latter formula is
xb = 0 which implies ab | xb trivially.
Otherwise b <r b1 therefore ab1 ≤ b. Then a1 b = 0, a1 <l a, b <r b1
implies ab1 = 0, hence b = 0. But then b <r b1 is not possible.
Now suppose n > 1. By Lemma 4.1 we may assume that n = 2 i.e.
a1 b1 | xb1 ∧ a2 b2 | xb2 → ab | xb. Let φi = ai bi | xbi , i = 1, 2 and let
φ = ab | xb. We may also assume that φi does not imply φ, in particular,
φ1 and φ2 are incomparable.
9
Thus, by symmetry, we may suppose that a2 <l a1 (if a1 ∼l a2 then φ1
and φ2 are obviously comparable). For the same reason one can assume that
b2 < r b 1 .
If a1 b1 ≤ a2 b2 then by Lemma 4.2 we will reduce the proof to the case
n = 1. Therefore let a1 b1 > a2 b2 :
O
a1 b1 | xb1
◦
a1 ◦
a2 ◦
a1 b1 > a2 b2
◦
a2 b2 | xb2
◦
◦
b1
◦
b2
/
By distributivity φ1 ∧ φ2 = (a1 | x + xb1 = 0) ∧ (a2 | x + xb2 = 0) = (a1 |
x ∧ a2 | x) + (a1 | x ∧ xb2 = 0) + (a2 | x ∧ xb1 = 0) + (xb1 = 0 ∧ xb2 = 0) =
a1 | x + (a2 | x ∧ xb1 = 0) + xb2 = 0.
Since this formula implies φ we obtain:
1) a1 | x → a | x + xb = 0;
2) a2 | x ∧ xb1 = 0 → a | x + xb = 0;
3) xb2 = 0 → a | x + xb = 0.
From 1) by Corollary 3.4 we obtain
a1 ∈ Ra i.e. a ≤l a1
or
a1 b = 0.
(∗)
Since φ1 does not imply φ and a1 | x → φ by 1), it follows that xb1 = 0
does not imply a | x + xb = 0. Then by Corollary 3.5 we obtain
Ra ⊂ Re,
b∈
/ b1 R i.e b <r b1
and
ab1 > b.
From 3), by Corollary 3.5, (keeping in mind that Ra ⊂ Re) we obtain
b ∈ b2 R i.e b2 ≤r b
or
ab2 ≤ b.
(∗ ∗ ∗)
Since φ2 does not imply φ and xb2 = 0 → φ by 3), it follows that a2 | x
does not imply a | x + xb = 0. Then, by Corollary 3.4, we obtain
a2 ∈
/ Ra i.e a2 <l a
and
a2 b ̸= 0.
Also from 2), by Lemma 3.1, (keeping in mind that a2 ∈
/ Ra, b ∈
/ b1 R) we
obtain
ab1 ≤ a2 b.
10
Let us return back to (∗). If a1 b = 0 and a1 ∈
/ Ra, then a ∈ Ra1 , therefore
ab = 0. Then b1 ∈ bR yields ab1 = 0. Then ab1 ≤ a2 b yields a2 b = 0, a
contradiction.
Thus by (∗)
a1 ∈ Ra
i.e.
a ≤l a1 .
Then a1 b1 | xb1 → ab1 | xb1 is a trivial implication. Let us return back to
(∗ ∗ ∗). If b ∈ b2 R then a2 b2 | xb2 → a2 b | xb trivially. Since ab1 ≤ a2 b, by
fusion ab1 | xb1 ∧ a2 b | xb is equivalent to ab | xb:
O
a1 b1 | xb1
◦
ab | xb
◦O o
◦
a1 ◦
a◦
/ ◦
◦
a2 b2 | xb2
/
◦
◦
◦
◦
b1
b2
b
Otherwise b ∈
/ b2 R therefore by (∗ ∗ ∗) we obtain ab2 ≤ b. Then b < ab1 ≤
a2 ◦
a2 b ≤ ab2 ≤ b, a contradiction.
We identify an e-type p(x) with the set of formulae ab | xb it contains.
Also, we identify each of these formulae with the corresponding point (b, a)
on the plane eR × Re. Let I = I(p) = {b ∈ eR | xb = 0 ∈ p} and
J = J(p) = {a ∈ Re such that ab | xb ∈ p for b ∈ eR implies xb = 0 ∈ p}. It
is easy to check that I ⊆ eR is a right ideal of R and J ⊆ Re is a left ideal
of R:
0 O
p−
J
◦
p−
.
. p+
p+
e ◦
e
Set I ∗ = eR \ I, J ∗ = Re \ J.
◦
I
/
0
Lemma 4.4. Let p be an e-type over a serial ring R, I = I(p) and J = J(p).
Then s∗ r∗ < sr for all s ∈ J, s∗ ∈ J ∗ , r ∈ I, r∗ ∈ I ∗ .
Proof. Otherwise, we obtain sr ∼ s∗ r∗ (since sr ∈ Rs∗ r∗ R). We may assume
that s∗ r∗ | xr∗ ∈ p. Indeed, otherwise s∗ r∗ | xr∗ ∈ p− . Since s∗ ∈ J ∗ , there
11
is r′ ∈ I ∗ such that s∗ r′ | xr′ ∈ p. Then clearly r′ ∈ r∗ R, s∗ r′ ∼ sr, and we
may replace r∗ by r′ .
Thus s∗ r∗ | xr∗ ∈ p. Since r ∈ I, it follows that s∗ r∗ | xr∗ ∧ sr | xr ∈ p.
By Lemma 4.2 (i.e by fusion), this formula is equivalent to sr∗ | xr∗ , which
yields sr∗ | xr∗ ∈ p. Since r∗ ∈ I ∗ , the definition of J yields s ∈
/ J, a
contradiction.
Recall that a pp-type p is called indecomposable if its pure-injective envelope N (p) is an indecomposable (pure-injective) module. Over a serial
ring, by [7, L. 11.6], an e-type is indecomposable iff from ab | xb ∈ p for
a ∈ Re, b ∈ eR, it follows that either a | x ∈ p or xb = 0 ∈ p. Thus an
indecomposable e-type p looks like a one-step ladder:
O
◦
J
. p+
p+
◦
◦
I
/
In particular, p is completely determined by the pair (J, I). Conversely,
given a left ideal J ⊆ Re and a right ideal I ⊆ eR, we may define an e-type
p = p(J ∗ /I) in the following way. For a ∈ Re, b ∈ eR set ab | xb ∈ p iff a ∈ J ∗
or b ∈ I. If p is consistent, it is indecomposable, and all indecomposable
e-types can be obtained in this way.
Corollary 4.5. The pp-type p(J ∗ /I) is consistent iff s∗ r∗ < sr for all r ∈ I,
r∗ ∈ I ∗ , s ∈ J, s∗ ∈ J ∗ .
5. Super-decomposable pure-injective modules
Recall that a module M is super-decomposable if it fails to have an indecomposable direct summand, and a pp-type p is called super-decomposable
if its pure-injective envelope N (p) is super-decomposable. Let p be a pptype. A pp-formula φ ∈ p− is large in p if for any φ1 , φ2 ∈ p− such that
φ → φ1 , φ2 , there is ψ ∈ p such that (φ1 ∧ ψ) + (φ2 ∧ ψ) ∈ p− . Note that if
the relevant part of the lattice of pp-formulae is distributive, then the latter
condition can be replaced by φ1 + φ2 ∈ p− . By [5, Thm. 9.16], a pp-type p
is super-decomposable iff p contains no large formula.
Lemma 5.1. Let p be a pp-type such that φ ∈ p and p contains no large
formula below φ. Then p is super-decomposable.
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Proof. Let m ∈ N (p) realize p, and assume that N (p) has an indecomposable direct summand. Then N (p) = N ⊕ K, where N is indecomposable.
Decompose m = n+k, n ∈ N , k ∈ K, in particular n, k ̸= 0. Since n ∈ N (p),
there is a formula ψ(x, y) such that N (p) |= ψ(m, n) and N (p) |= ¬ ψ(m, 0).
Forming the conjunction with φ(x), we may assume that ψ(x, 0) → φ(x).
Let θ(x) = ψ(x, 0). Projecting ψ(m, n) onto K, we obtain K |= θ(k),
hence N (p) |= θ(k). Since N (p) |= ¬ θ(m), it follows that N |= ¬ θ(n). Now
it is easy to see that p and q = ppN (n) are associated over θ, i.e. for every
pp-formula ξ with θ → ξ, ξ ∈ p iff ξ ∈ q. Since N is indecomposable, θ is
large in p and θ → φ, a contradiction.
Now we are in a position to describe serial rings with a super-decomposable
pure-injective module.
Theorem 5.2. Let R be a serial ring. Then the following are equivalent:
1) there exists a super-decomposable pure-injective module over R;
2) there exists an idempotent e ∈ R (which may be chosen from the original list e1 , . . . , en ) and embeddings of linear orders f : Q → Re, g : Q → eR
such that f (q) <l f (q ′ ), g(q) <r g(q ′ ) and f (q)g(q) < f (q ′ )g(q ′ ) for all
q < q ′ ∈ Q.
Proof. 1) ⇒ 2). Let M be a super-decomposable pure-injective module over
R and 0 ̸= m ∈ M . Then me ̸= 0 for some indecomposable idempotent
e ∈ R (clearly e can be chosen from the list e1 , . . . , en ). Thus the e-type
p = ppM (me) is super-decomposable. By [7, Fact 12.9] p contains no large
formula. Set I = I(p), J = J(p).
Step 1. There is ab | xb ∈ p− such that a ∈ J ∗ , b ∈ I ∗ .
Since the e-type p is nonzero, the zero formula φ = ‘0 | x + xe = 0’ is in
p− .
Since φ is not large in p, there are formulae φi = ai bi | xbi , ai ∈ Re,
bi ∈ eR, i = 1, 2 such that φ → φ1 , φ2 and φ1 + φ2 ∈ p. Then φ1 and φ2 are
incomparable, therefore we may assume that a1 <l a2 and b1 <r b2 which
yields φ1 + φ2 = a1 b2 | xb2 . Since a2 b2 | xb2 ∈ p− , it follows that b2 ∈ I ∗ ,
therefore b1 ∈ I ∗ . Also b2 ∈ I ∗ and a1 b2 | xb2 ∈ p imply a1 ∈ J ∗ .
Step 2. For any formula φ = ab | xb ∈ p− , a ∈ J ∗ , b ∈ I ∗ there are
a′ ∈ J ∗ , a′ <l a and b′ ∈ I ∗ , b′ >r b such that ab′ | xb′ , a′ b | xb ∈ p− and
a′ b′ | xb′ ∈ p:
13
O
J
◦ _ _ _ _ _ _ _ _ _ _ .
p−
p−
a◦
◦
◦
a′ ◦
p−
◦
◦
• +
p
◦
b
◦
b′
I
◦
/
Since φ is not large in p there are pp-formulae φ1 = a1 b1 | xb1 , φ2 =
a2 b2 | xb2 in p− such that φ → φ1 , φ2 and φ1 + φ2 ∈ p. Also we may assume
that a1 <l a2 , b1 <r b2 . Since a2 b2 | xb2 ∈ p− as above we obtain b1 , b2 ∈ I ∗ .
Since a ∈ J ∗ , Lemma 4.4 gives ab1 , ab2 ̸= 0. From φ → φ1 , φ2 by Lemma 3.6
we obtain a1 , a2 ≤ a, therefore a1 , a2 ∈ J ∗ .
If b ≤r b1 , b2 then the required formulae are ab2 | xb2 and a1 b | xb:
O
J
◦ _ _ _ _ _ _ _ _ _ _ _ .
a◦
◦
◦
a2 ◦
◦
a1 ◦
◦
◦
•
◦
◦
b
◦
b1
◦
b2
◦
I
/
For instance, since b ≤r b1 , we have a1 b | xb → a1 b1 | xb1 ∈ p− . Then
a1 b | xb ∈ p− .
Otherwise b1 <r b. Since ab | xb → a1 b1 | xb1 , Lemma 3.6 implies
a1 b ≤ b1 . If b2 ≤r b then a1 b2 ≤ b1 yields a2 b2 | xb2 → a1 b1 | xb1 , a
contradiction. Thus b <r b2 . Therefore we have obtained the following
configuration:
O
a◦
◦
◦O
a2 ◦
◦
a1 ◦
◦
/ ◦
•
◦
◦
b1
◦
b
◦
b2
14
/
Clearly ab2 | xb2 ∈ p− and ab2 | xb2 + a1 b | xb = a1 b2 | xb2 ∈ p. It remains
to prove that a1 b | xb ∈ p− . Otherwise a1 b | xb ∈ p. But since a1 b ≤ b1 then
(fusing with b1 | xb1 ) we obtain a1 b1 | xb1 ∈ p, a contradiction.
Continuing this construction we obtain a continually refined configuration:
O
◦
c3 ◦
c2 ◦
◦
◦
c1 ◦
◦
•
◦
◦
d1
◦
◦
◦
•
•
◦
d2
/
◦
d3
Let us show how to find the required maps from Q to Re and eR. We
check for instance, that c1 d1 < c2 d2 . Otherwise c1 d1 ∼ c2 d2 . Then by
Lemma 4.2, by fusion, we obtain c2 d1 | xd1 ∈ p, a contradiction. Thus
c1 d1 < c2 d2 < c3 d3 , and so on.
For instance, f (i) = ci , g(i) = di , i = 1, 2, 3 define maps from the three
element chain 1, 2, 3 ⊆ Q to Re and eR. To extend this map to 3/2 ∈ Q
(which is between 1 and 2) we use (d1 , c2 ) as a ‘root’ for the next step of
the construction.
2) ⇒ 1). Let us define the set of pp-formulae and their negations p =
p+ ∪ ¬ p− in the following way. For a ∈ Re, b ∈ eR, we put ab | xb ∈ p− if
there are q > q ′ ∈ Q such that f (q) ≤l a and b ≤r g(q ′ ). Otherwise we put
ab | xb ∈ p+ . Note that p has ‘block diagonal’ form:
O
◦
f (q) ◦
J
. p+
p−
•
p+
◦
◦
g(q)
◦
I
/
For instance, f (q)g(q) | xg(q) ∈ p for every q ∈ Q, and I(p) = {b ∈ eR |
b > g(q) for all q ∈ Q}.
15
Let us check that p is consistent. By Proposition 4.1 it suffices to check
that p is closed with respect to trivial implications and fusions.
Look first at trivial implications. Let ab | xb ∈ p+ → abd | xbd ∈ p− .
Then there are q > q ′ ∈ Q such that f (q) ≤l a and bd ≤r g(q ′ ). Since
b ≤r bd, we have b ≤ g(q ′ ), which yields ab | xb ∈ p− , a contradiction.
Trivial implications of the form ab | xb → cb | xb, a ≥l c are dealt with
similarly.
Now we consider fusions. Let φi = ai bi | xbi ∈ p, ai ∈ Re, bi ∈ eR,
i = 1, 2 be such that a1 <l a2 , b1 <r b2 and a1 b1 ∼ a2 b2 , i.e. φ1 ∧ φ2 is
equivalent to a2 b1 | xb1 by fusion. Let us assume that the latter formula is
in p− , i.e. that there are q > q ′ ∈ Q such that f (q) ≤l a2 and b1 ≤r g(q ′ ).
Choose q1 , q2 ∈ Q such that q ′ < q1 < q2 < q. We have b1 ≤r g(q ′ ) <r
g(q1 ), g(q2 ). Since a1 b1 | xb1 ∈ p, q1 > q2 and b1 ≤r g(q2 ), it follows that
a1 <l f (q1 ), which implies a1 b1 ≤ f (q1 )g(q1 ).
We also have f (q1 ), f (q2 ) <l f (q) ≤l a2 . Since a2 b2 | xb2 ∈ p, q1 > q2 and
f (q1 ) ≤l a2 , it follows that b2 ≥r g(q2 ), which implies a2 b2 ≥ f (q2 )g(q2 ).
Thus we obtain a1 b1 ≤ f (q1 )g(q1 ) < f (q2 )g(q2 ) ≤ a2 b2 , a contradiction.
Thus p is consistent. Now we prove that p is super-decomposable. By
Lemma 5.1 it suffices to prove that p contains no large formula below e | x.
Otherwise ab | xb ∈ p− , for some a ∈ Re, b ∈ eR, is large in p. Since
ab | xb ∈ p− , there are q > q ′ ∈ Q such that f (q) ≤l a and b ≤r g(q ′ ). Take
q1 ∈ Q such that q > q1 > q ′ .
O
p−
◦
a◦
f (q) ◦
f (q1 )◦
f (q ′ ) ◦
◦
◦
p−
•
p+
◦
b
◦
◦
g(q ′ ) g(q1 )
◦
g(q)
/
By the definition of p, since q1 > q ′ , we obtain f (q1 )b | xb ∈ p− . From
q > q1 it follows that ag(q1 ) | xg(q1 ) ∈ p− . Clearly ab | xb → ag(q1 ) | xg(q1 )
and f (q1 )b | xb. Also the sum of these formulae is f (q1 )g(q1 ) | xg(q1 ) ∈ p, a
contradiction.
16
6. Conclusions
Is the property of having a super-decomposable pure-injective module
left-right symmetric? In general it is known that this property is symmetric
for countable rings.
Corollary 6.1. Let R be a serial ring. If there exists a super-decomposable
pure-injective right module over R, then there exists a super-decomposable
pure-injective left module over R.
Proof. It is clear that condition 2) in Theorem 5.2 is left-right symmetric.
Reynders [9, Cor. 3.11] proved that any serial ring R with Krull dimension
(in the sense of Gabriel–Rentschler) fails to have a super-decomposable pureinjective module. The following result is somewhat stronger.
We say that a ring R has weak Krull dimension if the lattice of two-sided
ideals of R has Krull dimension. Thus R fails to have weak Krull dimension
iff the lattice of two-sided ideals of R contains a subset with the ordering of
the rationals.
Corollary 6.2. Let R be a serial ring with weak Krull dimension. Then
there is no super-decomposable pure-injective module over R.
Proof. Suppose that a super-decomposable pure-injective module over R
exists. Then by Theorem 5.2 we obtain a densely ordered set Rf (q)g(q)R,
q ∈ Q of two-sided ideals of R, a contradiction.
Nevertheless we do not know the answer to the following question.
Question 6.3. Let R be a serial ring such that the weak Krull dimension of
R is undefined. Does there exist a super-decomposable pure-injective module
over R?
By [5, Th. 10.9, 10.13], if R is a ring with a super-decomposable pureinjective module, then the lattice of all pp-formulae over R does not have
width. The converse has been proved only for countable rings.
For serial rings the situation is better.
Proposition 6.4. Let R be a serial ring. Then the following are equivalent:
1) there exists a super-decomposable pure-injective module over R;
2) the width of the lattice L of all pp-formulae over R is undefined.
17
Proof. We have just mentioned that 1) ⇒ 2) is true for any ring.
2) ⇒ 1). Let R be a serial ring such that L does not have width. By the
definition of width, L contains a countable dense ‘diamond’ sublattice (see
[5, Ch. 10]). It is quite easy to construct a countable elementary subring
R′ of R such that the lattice L′ of all pp-formulae over R′ does not have
width (we should include in R′ all the elements of R which occur in this
dense subset and also all elements witnessing implications between formulae
in this subset).
It is clear that R′ is serial if we include all the idempotents ei in R′ . Since
L′ does not have width and R′ is countable, there is a super-decomposable
pure-injective module over R′ .
Then Theorem 5.2 yields that for some i there are functions f : Q → R′ ei
and g : Q → ei R′ which satisfy condition 2) of this theorem. Since R′ is
an elementary subring of R, using the same functions we verify 2) over R.
Applying this theorem again, we obtain a super-decomposable pure-injective
module over R.
The author is indebted to Mike Prest and Philipp Rothmaler for helpful
comments.
References
[1] P. Eklof, I. Herzog, Model theory over a serial ring, Ann. Pure Appl. Math., 72
(1995), 145–176.
[2] A. Facchini, Module Theory: Endomorphism rings and direct sum decompositions in
some classes of modules. Birkhäuser, Progress in Mathematics, 167 (1998)
[3] G. Grätzer, General Lattice Theory. Academie Verlag, 1978.
[4] C.U. Jensen, H. Lenzing, Model Theoretic Algebra. Gordon and Breach, Algebra,
Logic and Applications, 2 (1989)
[5] M. Prest, Model Theory and Modules. Cambridge University Press, London Math.
Soc. Lecture Note Series, 130 (1987)
[6] M. Prest, G. Puninski, Some model theory over hereditary noetherian domains, J.
Algebra, 211 (1999), 268–297.
[7] G. Puninski, Serial Rings. Kluewer Academic Publishers, 2001.
[8] G. Puninski, Super-decomposable pure-injective modules exist over some string algebras, Preprint, 2001.
[9] G. Reynders, Ziegler spectra of a serial ring with Krull dimension, Comm. Algebra,
27(6) (1999), 2583–2611.
[10] J. Trlifaj, Two problems of Ziegler and uniform modules over regular rings, pp. 373–
383 in: Abelian Groups and Modules, Lect. Notes Pure Appl. Math., 182 (1996)
18
Department of Mathematics, The Ohio State University at Lima, 435 Galvin
Hall, 4240 Campus Drive, Lima Ohio 45804, USA
E-mail address: [email protected]
19