On Golomb Topology of Modules over Commutative Rings

Abstract

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 $\widetilde{G(M)}.$ 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.

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