On Golomb Topology of Modules over Commutative Rings

In this paper, we associate a new topology to a nonzero unital module $M$ over a commutative $R$, which is called Golomb topology of the $R$-module $M$. Let M\be an\ $R$-module and $B_M$ be the family of coprime cosets $\{m+N\}$ where $m\in M$ and N\is a nonzero submodule of M\such that $N+Rm=M$. We prove that if M\is a meet irreducible multiplication module or M\is a meet irreducible finitely generated module in which every maximal submodule is strongly irreducible, then B_{M}\is the basis for a topology on M\which is denoted by $\widetilde{G(M)}\mathrm{.}$ In particular, the subspace topology on $M-\{0\}$ is called the Golomb topology of the $R$-module M\and denoted by $G(M)$. We investigate the relations between topological properties of G(M)\and algebraic properties of M.\In particular, we characterize some important classes of modules such as simple modules, Jacobson semisimple modules in terms of Golomb topology.