Zariski Topologies for Coprime and Second Submodules

Abstract

Let $M$ be a non-zero module over an associative (not necessarily commutative) ring. In this paper, we investigate the so-called \emph{second} and \emph{coprime} submodules of $M.$ Moreover, we topologize the spectrum $% \mathrm{Spec}^{\mathrm{s}}(M)$ of second submodules of $M$ and the spectrum $% \mathrm{Spec}^{\mathrm{c}}(M)$ of coprime submodules of $M,$ study several properties of these spaces and investigate their interplay with the algebraic properties of $M.$