NORMSETS OF ALMOST DEDEKIND DOMIANS AND
ATOMICITY
RICHARD ERWIN HASENAUER
Abstract. Let D be an almost Dedekind domain. Then for each maximal ideal
M , we have DM being a Noetherian valuation domain. Using these local valuations
we create a global norm. The set of these norms, called the normset, is used to
classify which almost Dedekind domains are atomic. We also introduce the idea of
a legitimate domain and show that an atomic legitimate domain must have a zero
Jacobson radical. As a corollary, we show that a glad domain is atomic if and only
if it is a semilocal PID.
Let D be an almost Dedekind domain with the set of maximal ideals Max(D).
Now we have for all M ∈ Max(D), DM being a discrete valuation domain. Thus
for M ∈ Max(D) we have a valuation map νM : DM → Z. We also know that
D = ∩M ∈Max(D) DM . Thus a ∈ D if and only if νM (a) ≥ 0 for all M ∈ Max(D). We
piece together these local valuations to create a global valuation in the natural way.
That is we define the map
Y
N :D→
N0
M ∈Max(D)
by
N (a) = (νM (a))M ∈Max(D)
We will abuse terminology and refer to this net (sequence if Max(D) is countable) ,
as the norm of an element and the map as the norm map on D.
Let us make a few observations about N . First we see that N (ab) = N (a)+N (b) for
any elements a and b of D. This follows directly from the property of valuation maps.
Secondly we observe that if u is a unit in D, then νM (u) = 0 for all M ∈ Max(D).
We will say N (u) = 0 or N (u) is the zero net. Thirdly we make an observation that
will play a critical role in the way we start thinking of atoms in D. We shall say
N (a) ≤ N (b) if νM (a) ≤ νM (b) for all M ∈ Max(D). We will say N (a) < N (b) if
N (a) ≤ N (b) and νM (a) < νM (b) for some M ∈ Max(D). Lastly we will rely heavily
on the fact that νM (a + b) = min{νM (a), νM (b)} whenever νM (a) 6= νM (b).
Lemma 0.1. N (a) < N (b) if and only if a divides b.
Proof. Suppose N (a) < N (b). We have ab is in the quotient field of D. Now νM ( ab ) =
νM (b) − νM (a) ≥ 0 for all M ∈ Max(D). Thus ab ∈ DM for all M . Hence ab ∈ D. We
conclude that a divides b.
Suppose a divides b. Then ab ∈ D. Thus ab ∈ DM for all M ∈ Max(D). Thus for all
M we have νM ( ab ) ≥ 0. Hence νM (b) > νM (a) and we concluce that N (a) < N (b). We wish to only consider proper divisors. The lemma could be restated with
N (a) ≤ N (b), but in this case we might end up finding an associate, rather than a
proper divisor.
This lemma will help us find divisors. It should also be pointed out that this is not
true for the traditional norm in a Dedekind domain. To see this let us consider an
example.
Date: September 19, 2011.
√
Example
0.2. Consider D = Z[ −14].
The traditional Dedekind norm is N 0 (a +
√
√
2
2
0
0
b −14)
√ = a + 14b . Now N (5 + 2 −14) = 81 and N (3) = 9 but 3 does not divide
5 + 2 −14. But what√about our norm?
√
Let M1 = (3, 5 + 2 −14) and M2 = (3, 5 − 2 −14) It is easy to see that M1 and
M2 are maximal
in D and 3 ∈ M1√
, M2 . That is we have√νM1 (3) = 1 and νM2 (3) = 1.
√
Now 5√+ 2 −14 ∈ M
/ M2 . If 5 + 2 −14 ∈ M2 , we would have
√1 , but 5 + 2 −14 ∈
5 + 2 −14 + 5 − 2 −14 = 10 √∈ M2 but 3 and 10 are coprime in D.
√ Thus we
would have 1 ∈ M2 . Thus 5 + 2 √
−14 ∈
/ M2 . Thus we have νM2 (5 + 2 −14) = 0.
We conclude that
N
(3)
<
6
N
(5
+
2
−14).
Thus our norm recognizes that 3 is not a
√
divisor of 5 + 2 −14.
The next lemma will lead to a powerful observation that will help us view when an
almost Dedekind domain is atomic. We define the norm set of D as
Norm(D) = {N (a)| a ∈ D}
We observe that Norm(D) is a monoid under addition. The identity element is the
net of all zeros, which is in Norm(D) since D is assumed to be unital. As observed
before addition in the moniod translates to multiplication in the domain.
Lemma 0.3. N (a) is an atom in Norm(D) if and only if a is an atom in D.
Proof. Suppose N (a) is an atom in Norm(D). Suppose further a = bc ∈ D. Then
N (a) = N (b) + N (c) thus N (b) or N (c) must be the zero net. Thus b or c must be a
unit, and a is an atom.
Conversly, suppose a is atom in D. Suppose further N (a) = N (b) + N (c). Now if
both N (b) and N (c) are nonzero, then a = ubc is a proper factoring for some unit u.
Hence a is not an atom. Thus N (a) must be an atom in Norm(D).
We now arrive at a theorem that is in the spirit of [1].
Theorem 0.4. Let D be an almost Dedekind domain. D is atomic if and only if
Norm(D) is an atomic monoid.
Proof. Suppose D is atomic. Consider N (a). We have a = α1 α2 · · · αn a product of
atoms. Now we have N (a) = N (α1 ) + N (α2 ) + · · · + N (αn ) is an atomic factorization
in Norm(D).
Suppose Norm(D) is an atomic moniod. Consider a ∈ D. We write N (a) =
N (α1 ) + N (α2 ) + · · · + N (αn ) as an atomic factorization in Norm(D). We now see
a = uα1 α2 · · · αn for some unit u is an atomic factorization in D.
For convience we will say N (a) < ∞ or a has finite norm whenever a is only
contained in finitely many maximal ideals. That is the net N (a) has only finitely
many nonzero entries. Similarly we will say an element a has infinte norm if a is in
infinely many maximal ideals. We will write N (a) = ∞ in this case.
It should be noted that an element of finite norm can always be written as a product
of atoms. This is clear, for there are only finitely many possible divisors to choose
from, some of which must be atoms. Further an element of finite norm has only
finitely many possible atomic factorizations.
We will refer to an almost Dedekind domain that is not Dedekind as a purely almost
Dedekind domain. We now take a theorem from [2]
Theorem 0.5. In an integral domain D with identity which is not a field, the following conditions are equivalent:
(1) D is a Dedekind domain.
(2) D is almost Dedekind and each nonzero element of D is contained in only
finitely many maximal ideals in D.
Thus Dedekind domains contain only elements of finite norm, and a purely almost
Dedekind domain must contain an element of infinite norm.
We have a known fact as an obvious corollary.
Corollary 0.6. A Dedekind domain is a finite factorization domain.
We can characterize all the elements in the normset with the following theorem.
Theorem 0.7. Let D be an almost Dedekind domain with Max(D) = {Mλ }λ∈Λ . Then
\ e
eλ λ∈Λ ∈ Norm(D) ⇐⇒
Mλλ is a principal ideal.
λ∈Λ
0
Where we take M = D.
Proof. ⇒ Take a ∈ D with N (a) = (eλ )λ∈Λ . Recall νMλ (a) = eλ is equivalent to
/ Mλeλ +1 . Thus for all λ we have a ∈ Mλeλ , hence a ∈ ∩λ∈Λ Mλeλ .
saying a ∈ Mλeλ and a ∈
We will show this ideal is actually (a). Suppose b ∈ ∩λ∈Λ Mλeλ . Then for all λ we have
νMλ (b) ≥ eλ = νMλ (a). In other words we have N (a) ≤ N (b), hence a divides b. We
conclude that
\ eλ
Mλ .
a =
λ∈Λ
⇐ Suppose (a) =
∩λ∈Λ Mλeλ .
Then N (a) = (eλ )λ∈Λ .
We let max(b) = {M ∈ Max(D) | b ∈ M }.
Definition 0.8. We say a ∈ D is legitimate if N (a) is bounded. That is there exists
a ρ such that νM (a) < ρ for all M ∈ Max(D). A domain D is called legitimate if for
all a ∈ D, a is legitimate.
Theorem 0.9. If D is an atomicTpurely almost Dedekind domain that is legimate,
then the Jacobson radical J(D) = M ∈M ax(D) M = 0.
Proof. Since we are assuming that D is a purely almost Dedekind domain, then D
must have infinitely many primes. Suppose α 6= 0 ∈ J(D). Suppose α is an atom.
Now we note that ρ > νM (α) > 0 for all M and some fixed integer ρ. Now there
exists b ∈ D with b ∈
/ M 0 . We set ξ = bρ + α and note
νM (α) : M ∈ max(b)
νM (ξ) =
0
:M ∈
/ max(b)
Thus we see N (ξ) < N (α), hence ξ divides α. Hence α is not an atom.
If α is not an atom, we can write α = β1 β2 · · · βn as a product of atoms. Set
b = β2 β3 · · · βn , and note max(β1 ) ∪ max(b) = Max(D). We can assume max(b) 6=
Max(D). (If not we could consider b ∈ J(D), consider b0 = β3 · · · βn . If b0 ∈ J(D), we
continue until we have an atom in J(D).) We need to assure we create an element
that overlaps with β1 at some maximal ideal.
Claim: We can find M and M 0 in max(β1 ) \ max(b).
If max(β1 )\max(b) = P is a singleton set, then 1+b ∈ P since we are assuming b ∈
/
δ+1
J(D). Further 1+b is only in P . Now set νP (β1 ) = δ. Then we have νP (b +β1 ) = δ
and νM (bδ+1 + β1 ) = 0 for all other M . Hence bδ+1 + β1 divides β1 which is impossible.
Thus the claim must be true.
Now we find c ∈ M with c ∈
/ M 0 . Now we note that ρ > νM (β1 ) > 0 for all
M ∈ max(β1 ) and some fixed integer ρ. Set ξ = (bc)ρ + β1 . Thus we have

 νM (β1 ) : M ∈ max(bc) ∩ max(β1 )
0
: M ∈ max(bc) \ max(β1 )
νM (ξ) =
 0
: M ∈ max(β1 ) \ max(bc)
Thus we see N (ξ) < N (β1 ), hence ξ divides β1 . Thus β1 is not an atom. We conclude
that the intersection must be trivial.
Rephrasing we get the following corollary.
Corollary 0.10. An atomic legitimate domain with a nonzero Jacobson radical is a
semilocal Dedekind domain, hence semilocal PID.
Proof. If D is atomic, then D must have only finitely many primes. Thus it must be
Dedekind. Now semilocal Dedekind domains are semilocal PIDs.
In [3] Loper constructs a class of domains refered to as glad domains. In the
definition a glad domain is a legitimate domain. Rush shows in [5] that glad domains
have a nonzero Jacobson radical.
Corollary 0.11. A glad domain is atomic if and only if it is semilocal PID.
Another type of almost Dedekind domain that appears in the literature are SP domains. In [4] it is shown that SP -domains are legitimate.
Corollary 0.12. Let D be an SP -domain. If D is atomic then J(D) = 0 or D is a
semilocal PID.
References
[1] J. Coykendall, Normsets and determination of unique factorization in rings of algebraic integers.
Proc. Amer. Math. Soc.124(1996), no. 6, 1727-1732
[2] R. Gilmer, Multiplicative Ideal Theory Queen’s papers Pure Appl. Math. 90, Queen’s University
Press, Kingston 1992
[3] K.A. Loper, More almost Dedekind domains and Prúfer domains of polynomials, in Zerodimensional commutative rings (Knoxville, TN, 1994), 287-298, Lecture Notes in Pure and Appl.
Math., 171 (1995), Dekker, New York.
[4] B. Olberding, Factorization into radical ideals, Lec. Notes Pure Appl. 241 (2005), 363-377.
[5] D. Rush, The conditions Int(R)⊆ RS [X] and Int(RS ) = Int(R)S for integervalued polynomials,
Journal of Pure and Applied Algebra 125, (1998), 287-303.