Weakly S-Noetherian modules

Abstract

Let R be a commutative ring, S a multiplicative subset of R and M an R-module. We say that M satisfies weakly S-stationary on ascending chains of submodules (w-ACCS on submodules or weakly S-Noetherian) if for every ascending chain M1 ? M2 ? M3 ? ... of submodules of M, there exists k ? N such that for each n ? k, snMn ? Mk for some sn ? S. In this paper, we investigate modules (respectively, rings) with w-ACCS on submodules (respectively, ideals). We prove that if R satisfies w-ACCS on ideals, then R is a Goldie ring. Also, we prove that a semilocal commutative ring with w-ACCS on ideals have a finite number of minimal prime ideals. This extended a classical well known result of Noetherian rings.