Note on Colon-Multiplication Domains

Let R 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 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.


Introduction
Let R 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 colonmultiplication 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.

International Journal of Mathematics and Mathematical Sciences
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 Unreferenced material is standard, typically as in 1 or 2 .

Main Results
Definition 2.1.1 Let R be a domain, and A and B two nonzero fractional ideals of R. We 3 A domain R 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 domain R is said to be an MTP domain MTP stands for maximal trace property if for every nonzero fractional ideal For more details on the trace properties see 4 .
Theorem 2.2.Let R be an integral domain.The following statements are equivalent.
3 Either R is a Dedekind domain or a local MTP domain.
We need the following lemma.

Lemma 2.3. Let R be an integral domain and I a nonzero invertible (fractional) ideal of R. Then every nonzero (fractional) ideal
Proof.This follows immediately from the easily verified fact that if I is invertible, then A : I AI −1 for each nonzero ideal A.
Proof of Theorem 2.2. 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 , 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 .3 ⇒ 4 Suppose that R has a nonzero principal fractional ideal I aR that is colonmultiplication.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 ideal A 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-colonmultiplication.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 ideal M of a domain R which is colon-multiplication but not a multiplication ideal.
Let k be a field and X and Y indeterminates over 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 dim R 1 .By Theorem 2.2, M is colon-multiplication.However, M is not a multiplication ideal since M is not "locally" principal 5 .
2 Let R 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 ideal A of an integral domain, we define the map ϕ A : F R → F R , B → A B : A .The next proposition characterizes maps ϕ A that are surjective.Proposition 2.6.Let R be an integral domain and A a nonzero (fractional) ideal of R. The following conditions are equivalent.

1
ϕ A id (i.e., B is A-colon-multiplication for each B ∈ F R );2 ϕ A is surjective; 3 A is invertible.Proof. 1 ⇒ 2 Trivial. 2 ⇒ 3 Assume that ϕ A is surjective.Then there exists B ∈ F R such that A B : A ϕ A B R. Hence A is invertible.3 ⇒ 1 Assume that A is invertible.By Lemma 2.3, every B ∈ F R is A-colonmultiplication.Hence ϕ A BA B : A B and so ϕ A id.
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 −1The next result shows that colon-multiplication domains collapse to Dedekind domains.
R. Then necessarily AA −1 M. Hence A −1 M : A and therefore M AA −1 A M : A , as desired.