# Note on Colon-Multiplication Domains

Authors: A. Mimouni
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

LetR be an integral domain with quotient field L.  Call a nonzero (fractional) ideal A of R a colon-multiplication ideal any ideal A, such that B(A:B)=A
for every nonzero (fractional) ideal B of R. In this note, we characterize integral domains for which every maximal ideal (resp., every nonzero ideal) is a colon-multiplication ideal. It turns that this notion unifies Dedekind and MTP domains.


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## Body

## 1. Introduction

LetR be an integral domain which is not a field with quotient field L. For any nonzero (fractional) ideals A and B, B(A:B)⊆A and the inclusion may be strict. We say that A is B-colon-multiplication if equality holds, that is, A=B(A:B). A nonzero (fractional) ideal A is said to be a colon-multiplication ideal if A is B-colon-multiplication for every nonzero (fractional) ideal B of R, and the domain R is called a colon-multiplication domain if all its nonzero (fractional) ideals are colon-multiplication ideals. The purpose of this note is to characterize integral domains R that are colon-multiplication domains. This notion unifies the notions of Dedekind domains and MTP domains (i.e., domains R such that for every nonzero (fractional) ideal I, either I is invertible or II-1 is a maximal ideal of R). Precisely we prove that for a domain R, every maximal ideal is a colon-multiplication ideal if and only if either R is a Dedekind domain or a local MTP domain (Theorem 2.2), and a domain R is a colon-multiplication domain if and only if R is a Dedekind domain (Theorem 2.4). We also provide an example showing that the notions of colon-multiplication ideals and multiplication ideals (i.e., ideals A such that for every ideal B⊆A, there exists an ideal C such that B=AC) do not imply each other; however, over Noetherian domains, multiplication domains and colon-multiplication domains collapse to Dedekind domains.Throughout,R is an integral domain with quotient field L, Spec(R) denotes the set of all prime ideals of R, and F(R) denotes the set of all nonzero fractional ideals of R, that is, R-submodules of L such that dA⊆R for some nonzero d∈R. For A,B∈F(R), (A:B)={x∈L∣xB⊆A} and A-1=(R:A). Unreferenced material is standard, typically as in [1] or [2].


## 2. Main Results

Definition 2.1.
(1) LetR be a domain, and A and B two nonzero (fractional) ideals of R. We say that A is B-colon-multiplication if A=B(A:B).
(2) A nonzero (fractional) idealA is said to be a colon-multiplication ideal if A is B-colon-multiplication for every nonzero (fractional) ideal B of R.
(3) A domainR is said to be a colon-multiplication domain if every nonzero (fractional) ideal A of R is colon-multiplication.Our first main theorem characterizes integral domains for which every maximal ideal is colon-multiplication. Before stating the result, we recall that a domainR is said to be an MTP domain (MTP stands for maximal trace property) if for every nonzero (fractional) ideal I of R either II-1=R or II-1=M is a maximal ideal of R [3]. For more details on the trace properties see [4].Theorem 2.2.
LetR be an integral domain. The following statements are equivalent.(1)
Every nonzero prime ideal ofR is colon-multiplication;(2)
Every maximal ideal ofR is colon-multiplication;(3)
EitherR is a Dedekind domain or a local MTP domain.We need the following lemma.Lemma 2.3.
LetR be an integral domain and I a nonzero invertible (fractional) ideal of R. Then every nonzero (fractional) ideal A of R is I-colon-multiplication.Proof.
This follows immediately from the (easily verified) fact that ifI is invertible, then (A:I)=AI-1 for each nonzero ideal A.Proof of Theorem2.2.
(1)⇒ (2) Trivial.
(2)⇒ (3) First we claim that R is an MTP domain. Indeed, let I be a nonzero (fractional) ideal of R. Assume that II-1⊊R and let M be a maximal ideal such that II-1⊆M. Then I-1⊆(M:I)⊆I-1 and so I-1=(M:I). Since M is I-colon-multiplication, M=I(M:I)=II-1, and therefore R is an MTP domain. Now, if R is a Dedekind domain, we are done. Assume that R is not Dedekind. Then R is an MTP domain with a unique noninvertible maximal ideal M [4, Corollary 2.11]. Then MM-1=M. Now if N is a maximal ideal of R, by (2) N is M-colon-multiplication. So N=M(N:M)⊆MM-1=M and, by maximality, N=M. It follows that R is a local MTP domain, as desired.
(3)⇒ (1) If R is a Dedekind domain, then (1) it holds by Lemma 2.3. Assume that R is a local MTP domain. Then R is a one-dimensional domain [3, Proposition 2.10]. Hence Spec(R)={(0)⊊M} and so M is the unique nonzero prime ideal of R. Now, let A be a nonzero (fractional) ideal of R. If A is invertible, by Lemma 2.3,  M is A-colon-multiplication. Assume that AA-1⊊R. Then necessarily AA-1=M. Hence A-1=(M:A) and therefore M=AA-1=A(M:A), as desired.The next result shows that colon-multiplication domains collapse to Dedekind domains.Theorem 2.4.
LetR be an integral domain. The following statements are equivalent.(1)
R is a colon-multiplication domain;(2)
Every nonzero principal (fractional) ideal ofR is colon-multiplication;(3)
R has a nonzero principal (fractional) ideal that is colon-multiplication;(4)
R is a Dedekind domain.Proof.
(1)⇒(2)⇒(3) are trivial.
(3)⇒(4) Suppose that R has a nonzero principal (fractional) ideal I=aR that is colon-multiplication. Let J be any nonzero ideal of R. Then I is J-colon-multiplication. Hence aR=I=J(I:J)=J(aR:J)=aJJ-1 and therefore R=JJ-1, as desired.
(4)⇒(1). it Follows immediately from Lemma 2.3.We recall that an idealA of a commutative ring R is a multiplication ideal if for every ideal B⊆A there exists an ideal C such that B=AC, and the ring R is a multiplication ring if each ideal of R is a multiplication ideal. Note that from the equation B=AC, we have C⊆(B:A). Thus B=AC⊆A(B:A), and so we have B=A(B:A). Hence if A is a multiplication ideal of an integral domain R, then every subideal B of A is A-colon-multiplication. According to [5], a multiplication ideal is locally principal, but not conversely. However, a finitely generated locally principal ideal is a multiplication ideal [6]. In particular, in Noetherian domain, multiplication domain and colon-multiplication domain collapse to Dedekind domain. However, the two notions (multiplication and colon-multiplication) do not imply each other as is shown by the following example.Example 2.5.
(1) It provides a maximal idealM of a domain R which is colon-multiplication but not a multiplication ideal.
Letk be a field and X and Y indeterminates over k. Set R=k+Yk(X)[[Y]]=k+M. Clearly R is a one-dimensional PVD (pseudovaluation domain) and therefore a local MTP domain (here note that pseudovaluation domains have the trace property, [3, Example 2.12], and so the maximal trace property if dimR=1). By Theorem 2.2, M is colon-multiplication. However, M is not a multiplication ideal since M is not “locally” principal [5].
(2) LetR be a non-Dedekind domain. By Theorem 2.4, not every nonzero principal ideal is colon-multiplication. However, every principal ideal is a multiplication ideal [6].Given a nonzero (fractional) idealA of an integral domain, we define the map φA:F(R)→F(R),B↦A(B:A). The next proposition characterizes maps φA that are surjective.Proposition 2.6.
LetR be an integral domain and A a nonzero (fractional) ideal of R. The following conditions are equivalent.(1)
φA=id (i.e., B is A-colon-multiplication for each B∈F(R));(2)
φA is surjective;(3)
A is invertible.Proof.
(1)⇒(2) Trivial.
(2)⇒(3) Assume that φA is surjective. Then there exists B∈F(R) such that A(B:A)=φA(B)=R. Hence A is invertible.
(3)⇒(1) Assume that A is invertible. By Lemma 2.3, every B∈F(R) is A-colon-multiplication. Hence φA(B)=A(B:A)=B and so φA=id.



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