Korean J. Math. Vol. 28 No. 4 (2020) pp.907-913
DOI: https://doi.org/10.11568/kjm.2020.28.4.907

A characterization of $w$-Artinian modules

Main Article Content

Hwankoo Kim
Tae In Kwon
De Chuan Zhou

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.


Article Details

Supporting Agencies

Changwon National University

References

[1] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer- Verlag, New York, Berlin, 1992. Google Scholar

[2] C. Faith, Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Amer. Math. Soc, Providence, 1999. Google Scholar

[3] J.L. Garc ia Hern andez and J.L. Gomez Pardo, V-rings relative to Gabriel topolo- gies, Comm. Algebra 13 (1985), 59-83. Google Scholar

[4] J.P. Jans, Co-Noetherian rings, J. London Math. Soc. 1 (1) (1969). 588–590. Google Scholar

[5] A.J. P ena, Una caracterizaci on de los m odulos Artinianos, Divulg. Mat. 1 (1) (1993), 17-24. Google Scholar

[6] P. Vamos, The dual of the notion of finitely generated, J. London Math. Soc. 43 (1968), 643–646. Google Scholar

[7] F.G. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Singapore, Springer, 2016. Google Scholar

[8] H. Yin, F.G. Wang, X. Zhu, and Y. Chen, w-Modules over commutative rings, J. Korean Math. Soc. 48 (2011), 207–222. Google Scholar

[9] J. Zhang and F.G. Wang, The w-socles of w-modules, J. Sichuan Normal Univ. (Nat. Sci.) 36 (2013), 807–810. (in Chinese). Google Scholar

[10] D.C. Zhou, H. Kim, and K. Hu, A Cohen-type theorem for w-Artinian modules, J. Algebra Appl. 20 (2021), 2150106 (25 pages), DOI: 10.1142/S0219498821501061 Google Scholar