The Radical-Zariski topology on the radical spectrum of modules

Abstract

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