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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 231326, 4 pages
http://dx.doi.org/10.1155/2010/231326
Research Article

Note on Colon-Multiplication Domains

Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, P.O. Box 278, Dhahran 31261, Saudi Arabia

Received 8 April 2010; Accepted 25 August 2010

Academic Editor: David Dobbs

Copyright © 2010 A. Mimouni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in, any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

Let be an integral domain which is not a field with quotient field . For any nonzero (fractional) ideals and , and the inclusion may be strict. We say that is -colon-multiplication if equality holds, that is, . A nonzero (fractional) ideal is said to be a colon-multiplication ideal if is -colon-multiplication for every nonzero (fractional) ideal of , and the domain 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 that are colon-multiplication domains. This notion unifies the notions of Dedekind domains and domains (i.e., domains such that for every nonzero (fractional) ideal , either is invertible or is a maximal ideal of ). Precisely we prove that for a domain , every maximal ideal is a colon-multiplication ideal if and only if either is a Dedekind domain or a local domain (Theorem 2.2), and a domain is a colon-multiplication domain if and only if 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 such that for every ideal , there exists an ideal such that ) do not imply each other; however, over Noetherian domains, multiplication domains and colon-multiplication domains collapse to Dedekind domains.

Throughout, is an integral domain with quotient field , denotes the set of all prime ideals of and denotes the set of all nonzero fractional ideals of , that is, -submodules of such that for some nonzero . For , and . Unreferenced material is standard, typically as in [1] or [2].

2. Main Results

Definition 2.1. (1) Let be a domain, and and two nonzero (fractional) ideals of . We say that is -colon-multiplication if .
(2) A nonzero (fractional) ideal is said to be a colon-multiplication ideal if is -colon-multiplication for every nonzero (fractional) ideal of .
(3) A domain is said to be a colon-multiplication domain if every nonzero (fractional) ideal of 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 is said to be an domain ( stands for maximal trace property) if for every nonzero (fractional) ideal of either or is a maximal ideal of [3]. For more details on the trace properties see [4].

Theorem 2.2. Let be an integral domain. The following statements are equivalent.(1)Every nonzero prime ideal of is colon-multiplication;(2)Every maximal ideal of is colon-multiplication;(3)Either is a Dedekind domain or a local domain.

We need the following lemma.

Lemma 2.3. Let be an integral domain and a nonzero invertible (fractional) ideal of . Then every nonzero (fractional) ideal of is -colon-multiplication.

Proof. This follows immediately from the (easily verified) fact that if is invertible, then for each nonzero ideal .

Proof of Theorem 2.2. (1) (2) Trivial.
(2) (3) First we claim that is an domain. Indeed, let be a nonzero (fractional) ideal of . Assume that and let be a maximal ideal such that . Then and so . Since is -colon-multiplication, and therefore is an domain. Now, if is a Dedekind domain, we are done. Assume that is not Dedekind. Then is an domain with a unique noninvertible maximal ideal [4, Corollary ]. Then . Now if is a maximal ideal of , by (2) is -colon-multiplication. So and, by maximality, . It follows that is a local domain, as desired.
(3) (1) If is a Dedekind domain, then (1) it holds by Lemma 2.3. Assume that is a local domain. Then is a one-dimensional domain [3, Proposition ]. Hence and so is the unique nonzero prime ideal of . Now, let be a nonzero (fractional) ideal of . If is invertible, by Lemma 2.3, is -colon-multiplication. Assume that . Then necessarily . Hence and therefore , as desired.

The next result shows that colon-multiplication domains collapse to Dedekind domains.

Theorem 2.4. Let be an integral domain. The following statements are equivalent.(1) is a colon-multiplication domain;(2)Every nonzero principal (fractional) ideal of is colon-multiplication;(3) has a nonzero principal (fractional) ideal that is colon-multiplication;(4) is a Dedekind domain.

Proof. (1) (2) (3) are trivial.
(3) (4) Suppose that has a nonzero principal (fractional) ideal that is colon-multiplication. Let be any nonzero ideal of . Then is -colon-multiplication. Hence and therefore , as desired.
(4) (1). it Follows immediately from Lemma 2.3.

We recall that an ideal of a commutative ring is a multiplication ideal if for every ideal there exists an ideal such that , and the ring is a multiplication ring if each ideal of is a multiplication ideal. Note that from the equation , we have . Thus , and so we have . Hence if is a multiplication ideal of an integral domain , then every subideal of is -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 ideal of a domain which is colon-multiplication but not a multiplication ideal.
Let be a field and and indeterminates over . Set . Clearly is a one-dimensional (pseudovaluation domain) and therefore a local domain (here note that pseudovaluation domains have the trace property, [3, Example ], and so the maximal trace property if ). By Theorem 2.2, is colon-multiplication. However, is not a multiplication ideal since is not “locally” principal [5].
(2) Let 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 of an integral domain, we define the map . The next proposition characterizes maps that are surjective.

Proposition 2.6. Let be an integral domain and a nonzero (fractional) ideal of . The following conditions are equivalent.(1) (i.e., is -colon-multiplication for each );(2) is surjective;(3) is invertible.

Proof. (1) (2) Trivial.
(2) (3) Assume that is surjective. Then there exists such that . Hence is invertible.
(3) (1) Assume that is invertible. By Lemma 2.3, every is -colon-multiplication. Hence and so .

Acknowledgment

This work was supported by KFUPM.

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