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-ACC S on submodules or weakly S-Noetherian) if for every ascending chain M 1 ⊆ M 2 ⊆ M 3 ⊆ · · · of submodules of M, there exists k ∈ N such that for each n ≥ k, s n M n ⊆ M k for some s n ∈ S. In this paper, we investigate modules (respectively, rings) with w-ACC S on submodules (respectively, ideals). We prove that if R satisfies w-ACC S on ideals, then R is a Goldie ring. Also, we prove that a semilocal commutative ring with w-ACC S on ideals have a finite number of minimal prime ideals. This extended a classical well known result of Noetherian rings.