S-principal ideal multiplication modules

Abstract

In this paper, we study S-Principal ideal multiplication modules. Let A be a commutative ring with 1 &NOTEQUexpressionL; 0, S subset of A a multiplicatively closed set and M an A module. A submodule N of M is said to be an S-multiple of M if there exist s is an element of S and a principal ideal I of A such that sN subset of IM subset of N.M is said to be an S-principal ideal multiplication module if every submodule N of M is an S-multiple of M. Various examples and properties of S-principal ideal multiplication modules are given. We investigate the conditions under which the trivial extension Ax(SIC)M is an S x(SIC) 0-principal ideal ring. Also, we prove Cohen type theorem for S principal ideal multiplication modules in terms of S-prime submodules.