On modules satisfying S-dccr condition

Abstract

In this paper, we introduce a newclass ofmodules satisfying S-dccr (S-dccr*) condition which is a generalization of S-artinian modules. Let A be a commutative ring with 0 not equal 1 and X a unital A-module. Suppose that S subset of A is amultiplicatively closed subset. X is said to satisfy S-dccr ( S-dccr*) condition if for each finitely generated (principal) ideal I of A and a submodule Y of X, the descending chain {(IY)-Y-i}(i is an element of N) is S-stationary. Many examples and properties of modules satisfying S-dccr ( S-dccr*) condition are given. Furthermore, we characterizemodules satisfying dccr (dccr*) condition in terms of some known class of rings and modules. Also, we give Nakayama's Lemma for modules satisfying S-dccr condition.