Note on Colon-Multiplication Domains

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 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.

1. 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 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) Let 𝑅 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) ideal 𝐴 is said to be a colon-multiplication ideal if 𝐴 is 𝐵-colon-multiplication for every nonzero (fractional) ideal 𝐵 of 𝑅.
(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 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. Let 𝑅 be an integral domain. The following statements are equivalent.(1)Every nonzero prime ideal of 𝑅 is colon-multiplication;(2)Every maximal ideal of 𝑅 is colon-multiplication;(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 𝐴 of 𝑅 is 𝐼-colon-multiplication.

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 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. Let 𝑅 be an integral domain. The following statements are equivalent.(1)𝑅 is a colon-multiplication domain;(2)Every nonzero principal (fractional) ideal of 𝑅 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 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-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 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 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) 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-colon-multiplication. Hence 𝜑_𝐴(𝐵)=𝐴(𝐵∶𝐴)=𝐵 and so 𝜑_𝐴=𝑖𝑑.

Acknowledgment

This work was supported by KFUPM.