S-Artinian rings and finitely S-cogenerated rings

Abstract

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be a multiplicatively closed subset. In this paper, we study [Formula: see text]-Artinian rings and finitely [Formula: see text]-cogenerated rings. A commutative ring [Formula: see text] is said to be an [Formula: see text]-Artinian ring if for each descending chain of ideals [Formula: see text] of [Formula: see text] there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text] Also, [Formula: see text] is called a finitely [Formula: see text]-cogenerated ring if for each family of ideals [Formula: see text] of [Formula: see text] where [Formula: see text] is an index set, [Formula: see text] implies [Formula: see text] for some [Formula: see text] and a finite subset [Formula: see text] Moreover, we characterize some special rings such as Artinian rings and finitely cogenerated rings. Also, we extend many properties of Artinian rings and finitely cogenerated rings to [Formula: see text]-Artinian rings and finitely [Formula: see text]-cogenerated rings.

Weakly S-Artinian modules

“…Definition 2.2. According to [16], a commutative ring R is said to be an S-Artinian ring if for every descending chain of ideals of the form I 1 ⊇ I 2 ⊇ I 3 ⊇ • • • of R, there exist s ∈ S and k ∈ N such that sI k ⊆ I n for all n ≥ k. We can generalize this definition to noncommutative setting in a natural way as above.…”

“…Let R be a ring and S a multiplicative subset of R. It is well known that if R is an S-Artinian ring, then S −1 R is an Artinian ring ( [16]). Our next Proposition is obtained by relaxing the S-Artinian property.…”

“…They called a family F of submodules of a right R-module M, S-saturated if for every submodule N of M, whenever there exist s ∈ S and N 0 ∈ F such that Ns ⊆ N 0 , then N ∈ F. They proved that M is S-Noetherian if and only if every increasing sequence of submodules of M is S-stationary if and only if every nonempty set of submodules of M has an S-maximal element if and only if every nonempty S-saturated set of submodules of M has a maximal element. In [16], Sevim, Tekir and Koc studied the duality of the S-Noetherian concept. They introduced the S-Artinian rings and finitely S-cogenerated rings.…”

Characterizing $S$-projective modules and $S$-semisimple rings by uniformity

“…This is a well generation of Noetherian rings using multiplicative subsets. Since then, the notions of S-analogues of other well-known rings, such as artinian rings, coherent rings, almost perfect rings, GCD domains and strong Mori domains, are introduced and studied extensively in [2,4,5,9,10,12,15]. Now let's go back to the definition of S-Noetherian rings.…”

An extension of S-artinian rings and modules to a hereditary torsion theory setting

“…In [12], the author study S-artinian rings, dualizing the former notion of S-noetherian ring, and give some characterization of S-artinian rings in terms of finite cogeneration with respect to S. Our aim is to show that this theory is part of a more general theory involving hereditary torsion theories. In particular, we show that if A is totally σ-artinian, then the hereditary torsion theory σ is of finite type, and, in addition, A it is totally σnoetherian.…”