Weakly s-Artinian modules

Omid Khani-Nasab ; Ahmed Hamed

Filomat, Tome 35 (2021) no. 15, p. 5215

Résumé

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 N 1 ⊇ N 2 ⊇ N 3 ⊇ · · · of submodules of M is weakly S-stationary, i.e., there exists k ∈ N such that for each n ≥ k, s n N k ⊆ N n for some s n ∈ 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

Détail

DOI : 10.2298/FIL2115215K

Classification : 16P20, 16P70, 16U20
Keywords: weakly S-stationary, weakly S-Artinian modules, S-Artinian modules