HK-Space and Separation axioms of Spectra of maximal Sub modules

Abstract

The main purpose of this research, is to study the relationship between some the topological properties of the space Max(M) and the algebraic properties of an R-module M, and vice versa. We prove in this paper that if M is a multiplication R-module then M is semi local if and only if max(M) is an HK space. Also in our study we shall prove if M is a multiplication R-module and max(M) is an HK space then M is cyclic and we study Separation Axioms of the space maximal (M) which that we get the following results , If M is an M-top R-module and semi local then maximal (M) is T0 , T1 , T2 , T3 , T4 .