Some applications of Weak-multiplication modules in Neutrosophic set environment

Recently it has been proven that the multiplication module is a Weak-multiplication module (W −M M ). In this paper, we present some applications of W − M M in neutrosophy theory. n-refined neutrosophic module M denoted by Mn(I). We proved that Mn(I) is regular and C.P.module. Also, every n-refined neutrosophic finitely generated ideal is a principal ideal of n-refined the ring R (Rn(I)). We proved that ifMn(I) is multiplication module over ring Rn(I) and Mn(I) is of type S2; then every f. generated An(I) of Mn(I) is n-refined multiplication module. Also, if Rn(I) is n-refined neutrosophic ring, so Rn(I) is principal ideal ring if and only if every n-refined neutrosophic multiplication module Mn(I) is of type S. Finally, several results and applications have been presented in this paper with some new definitions, examples, and other properties.