S-Artinian rings and finitely S-cogenerated rings

Let R be a commutative ring with nonzero identity and S subset of R be a multiplicatively closed subset. In this paper, we study S-Artinian rings and finitely S-cogenerated rings. A commutative ring R is said to be an S-Artinian ring if for each descending chain of ideals {In}(n is an element of N) of R, there exist s is an element of S and k is an element of N such that sI(k) subset of I-n for all n >= k. Also, R is called a finitely S-cogenerated ring if for each family of ideals {I alpha}(alpha)(is an element of Delta) of R, = where Delta is an index set, boolean AND(alpha is an element of Delta) I alpha implies = 0 implies s(boolean AND(alpha is an element of Delta), I alpha) = 0 for some s is an element of S and a finite subset Delta' subset of Delta. 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 S-Artinian rings and finitely S-cogenerated rings.