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.