Revista de la
Unión Matemática Argentina
Uniformly $S$-Artinian rings and modules
Xiaolei Zhang and Wei Qi

Volume 69, no. 2 (2026), pp. 459–474    

Published online (final version): July 17, 2026

https://doi.org/10.33044/revuma.5558

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Abstract

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be a $u$-$S$-Artinian module if there is $s\in S$ such that any descending chain of submodules of $M$ is $S$-stationary with respect to $s$. The notion of $u$-$S$-Artinian modules is characterized in terms of ($S$-MIN)-conditions and $u$-$S$-cofinite properties. A ring $R$ is said to be $u$-$S$-Artinian if $R$ itself is a $u$-$S$-Artinian module. It is then shown that any $u$-$S$-semisimple ring is $u$-$S$-Artinian. It is proved that a ring $R$ is $u$-$S$-Artinian if and only if $R$ is $u$-$S$-Noetherian, the $u$-$S$-Jacobson radical $\operatorname{Jac}_S(R)$ of $R$ is $S$-nilpotent and $R/\operatorname{Jac}_S(R)$ is a $u$-$S/\operatorname{Jac}_S(R)$-semisimple ring. Besides, some examples are given to distinguish Artinian rings, $u$-$S$-Artinian rings, and $S$-Artinian rings.

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