Factorizations in Modules and Splitting Multiplicatively Closed Subsets

We introduce the concept of multiplicatively closed subsets of a commutative ring $R$ which split an $R$-module $M$ and study factorization properties of elements of $M$ with respect to such a set. Also we demonstrate how one can utilize this concept to investigate factorization properties of $R$ and deduce some Nagata type theorems relating factorization properties of $R$ to those of its localizations, when $R$ is an integral domain.