Geodesic
Weakly S-Noetherian modules
Filomat, Tome 37 (2023) no. 14, p. 4649

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DOI

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.
DOI : 10.2298/FIL2314649K
Classification : 13C, 13E05
Keywords: S-Noetherian ring, weakly S-stationary, weakly S-maximal 
Omid Khani-Nasab; Ahmed Hamed; Achraf Malek. Weakly S-Noetherian modules. Filomat, Tome 37 (2023) no. 14, p. 4649 . doi: 10.2298/FIL2314649K
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     title = {Weakly {S-Noetherian} modules},
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     year = {2023},
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     doi = {10.2298/FIL2314649K},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2314649K/}
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