S-Artinian rings and finitely S-cogenerated rings

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.