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) 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.