### Weakly S-Noetherian modules

#### Abstract

Let

an

of submodules (w-ACC

M2 ⊆ M3 ⊆ · · · of submodules of M , there exists k ∈ N such that for each

n ≥ k , snMn ⊆ Mk for some sn ∈

(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.

*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 chainsof submodules (w-ACC

*S*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.

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