A characterization of ω-Artinian modules
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Let $R$ be a commutative ring with identity and let $M$ be a $w$-module over $R$. Denote by $\mathscr{F}_M$ the set of all $w$-submodules of $M$ such that $(M/N)_w$ is $w$-cofinitely generated. Then it is shown that $M$ is $w$-Artinian if and only if $\mathscr{F}_M$ is closed under arbitrary intersections, if and only if $\mathscr{F}_M$ satisfies the descending chain condition.
Abstract
Let $R$ be a commutative ring with identity and let $M$ be a $w$-module over $R$. Denote by $\mathscr{F}_M$ the set of all $w$-submodules of $M$ such that $(M/N)_w$ is $w$-cofinitely generated. Then it is shown that $M$ is $w$-Artinian if and only if $\mathscr{F}_M$ is closed under arbitrary intersections, if and only if $\mathscr{F}_M$ satisfies the descending chain condition.
참고문헌 (10)
- F.W. Anderson, "Rings and Categories of Modules" (1992), SpringerVerlag
- C. Faith, "Rings and Things and a Fine Array of Twentieth Century Associative Algebra", Amer. Math. Soc (1999)
- J. L. Garc´ıa Hern´andez, "V-rings relative to Gabriel topologies", Comm. Algebra, 13, pp.59~83 (1985)
- J.P. Jans, "Co-Noetherian rings", J. London Math. Soc, 1(1), pp.588~590 (1969)
- A. J. Pena, "Una caracterizaci´on de los m´odulos Artinianos", Divulg. Mat, 1(1), pp.17~24 (1993)
- P. Vamos, "The dual of the notion of finitely generated", J. London Math. Soc, 43, pp.643~646 (1968)
- F.G. Wang, "Foundations of Commutative Rings and Their Modules" (2016), Springer
- Huayu Yin, "ω-MODULES OVER COMMUTATIVE RINGS", 대한수학회지, 48(1), pp.207~222 (2011), 대한수학회
- J. Zhang, "The w-socles of w-modules", J. Sichuan Normal Univ. (Nat. Sci.), 36, pp.807~810 (2013)
- D. C. Zhou, "A Cohen-type theorem for wArtinian modules", J. Algebra Appl, 20, pp.25~ (2021)[DOI]