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A Cohen-type theorem for $w$ -Artinian modules

Dechuan Zhou

Southwest University of Science and Technology, Mianyang 621010, P. R. China

E-mail Address: dechuan11119@sina.com

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Hwankoo Kim

Division of Computer & Information Engineering, Hoseo University, Asan 31499, Republic of Korea

E-mail Address: hkkim@hoseo.edu

Corresponding author.

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, and

Kui Hu

Southwest University of Science and Technology, Mianyang 621010, P. R. China

E-mail Address: hukui200418@163.com

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https://doi.org/10.1142/S0219498821501061Cited by:7 (Source: Crossref)

Abstract

Let $R$ be a commutative ring with identity. In this paper, a Cohen-type theorem for $w$ -Artinian modules is given, i.e. a $w$ -cofinitely generated $R$ -module $M$ is $w$ -Artinian if and only if ${(M / {ann}_{M} (P))}_{w}$ is $w$ -cofinitely generated for every prime $w$ -ideal $P$ of $R$ . As a by-product of the proof, we also obtain a detailed representation of elements of a $w$ -module and the $w$ -theoretic version of the Chinese remainder theorem for both modules and rings.

Communicated by A. Facchini

Keywords:

AMSC: 13E10, 13D30

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