On Multiplicative Subsets in Module Theory

Abstract

Let R be a commutative ring with identity. Multiplicative subset S in R (Mult- (S −1R)), means if 1∈S and ab∈S for all a, b ∈ S. In this paper, we characterize some particular cases of multiplicative subset in general and we studied the multiplicative subset by using two types of module as projective and flat modules. The main results about multiplicative subsets in module theory have relations with projective and flat module. We proved that if S is a Mult-(S −1R), so S-1R is a flat R-module. Also we obtained if S is a Mult-(S −1R) and P is a projective module, so S−1P is a projective S−1R-module. The other result is if S is a Mult-(S −1R), and M a module, then M=S−1M iff M is an S−1R-module. Finally we obtained the following result; if S is a multiplicative subset of a ring R, then S−1R=0 iff S contains a nilpotent element.