On uniformly $S$-Artinian rings and modules

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 uniformly S-Artinian (u-S-Artinian for abbreviation) module if there is s ∈ S such that any descending chain of submodules of M is S-stationary with respect to s. u-S-Artinian modules are characterized in terms of (S-MIN)-conditions and u-S-cofinite properties. We call a ring R is a u-S-Artinian ring if R itself is a u-S-Artinian module, and then show 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 Jac S (R) of R is S-nilpotent and R/Jac S (R) is a u-S/Jac S (R)-semisimple ring. Besides, some examples are given to distinguish Artinian rings, u-S-Artinian rings and S-Artinian rings.