Weakly S-Artinian Modules

Abstract

Let R be a ring, S a multiplicative subset of R and M a left Rmodule. We say M is a weakly S-Artinian module if every descending chain
N1 ⊇ N2 ⊇ N3 ⊇ · · · of submodules of M is weakly S-stationary, i.e., there
exists k ∈ N such that for each n ≥ k, snNk ⊆ Nn for some sn ∈ S. One
aim of this paper is to study the class of such modules. We show that over
an integral domain, weakly S-Artinian forces S to be R \ {0}, whenever S
is a saturated multiplicative set. Also we investigate conditions under which
weakly S-Artinian implies Artinian. In the second part of this paper, we
focus on multiplicative sets with no zero divisors. We show that with such
a multiplicative set, a semiprime ring with weakly S-Artinian on left ideals
and essential left socle is semisimple Artinian. Finally, we close the paper by
showing that over a perfect ring weakly S-Artinian and Artinian are equivalent.