The radical-Zariski topology on the radical spectrum of a module

Abstract

For a module $M$ over a commutative ring $R$ with identity‎, ‎let $\operatorname{RSpec}(M)$ denote the collection of all submodules $L$ of $M$ such that $\sqrt{(L:M)}$ is a prime ideal of $R$ and is equal to $(\operatorname{rad} L:M)$‎. ‎In this article‎, ‎we topologies $\operatorname{RSpec}(M)$ with a topology which enjoys analogs of many of the properties of the Zariski topology on the prime spectrum $\operatorname{Spec}(M)$‎. ‎We investigate this topological space from the point of view of spectral spaces by establishing interrelations between $\operatorname{RSpec}(M)$ and $\operatorname{Spec}(R/\ann(M))$‎.