Weakly S-Artinian modules

Let R be a ring, S a multiplicative subset of R and M a left R-module. 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 n f0g; 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.