On Rings and Modules with S - a:c:c

In this article we introduce and study S-a.c.c rings (modules) and S- nitely generated rings (modules)where S is a multiplicatively closed subset of a ring R and shows that a ring R is Noetherian if and onlyif, is P -a.c.c for all prime ideals of R if and only if, is M -a.c.c for all maximal ideals M of it and provesome relations between the concepts S- a.c.c and S- nitely generated.

کلیدواژه ها:

S - a.c.c ring ، S - a.c.c module ، S-f.g ring ، S-f.g module