A Cohen-type theorem for w-Artinian modules

Abstract

Let [Formula: see text] be a commutative ring with identity. In this paper, a Cohen-type theorem for [Formula: see text]-Artinian modules is given, i.e. a [Formula: see text]-cofinitely generated [Formula: see text]-module [Formula: see text] is [Formula: see text]-Artinian if and only if [Formula: see text] is [Formula: see text]-cofinitely generated for every prime [Formula: see text]-ideal [Formula: see text] of [Formula: see text]. As a by-product of the proof, we also obtain a detailed representation of elements of a [Formula: see text]-module and the [Formula: see text]-theoretic version of the Chinese remainder theorem for both modules and rings.

A note on a Cohen-type theorem for $w$-Artinian modules

“…In this note, we gave a new Cohen-type Theorem for w-Artinian modules which generalizes [15,Theorem 4.10] and [14,Theorem 2.1]. Actually, we obtain the following main result of this note.…”

“…Lemma 2.3. [15,Proposition 2.10] The following statements are equivalent for a GV-torsion-free R-module M.…”

“…Then N p is a w-submodule of M by Lemma 2.1. It follows by [15,Theorem 4.10] that (M/N p ) w is w-cofinitely generated with M[p] ⊆ N p ⊆ (0 : M p).…”

“…To extend the notion of w-Artinian modules, Zhou, Kim and Hu [15,Definition 2.1] called a w-module M w-cofinitely generated if for any set {M i |i ∈ Ω} of wsubmodules of M satisfying i∈Ω M i = 0, there exists a finite subset Ω 0 ⊆ Ω such that i∈Ω 0 M i = 0. They showed that a w-module M is w-cofinitely generated if and only if it is an essential extension of a w-Artinian module, if and only if every inverse system of nonzero w-submodules of M is bounded below by a nonzero w-submodule of M (see [15,Theorem 2.4,Proposition 2.11]). And finally, they obtained a Cohen-type Theorem for w-Artinian modules, which can be seen as a w-analogue of Nishitani's result in [7]: Theorem 1.2.…”

“…And finally, they obtained a Cohen-type Theorem for w-Artinian modules, which can be seen as a w-analogue of Nishitani's result in [7]: Theorem 1.2. [15,Theorem 4.10] Let R be a ring. A w-module M over R is w-Artinian if and only if M is w-cofinitely generated and (M/(0 : M p)) w is w-cofinitely generated for every prime w-ideal p of R.…”