Modules and the Second Classical Zariski Topology

Abstract

Let R be an associative ring with identity and Spec^{s}(M) denote the set of all second submodules of a right R-module M. In this paper, we present a number of new results for the second classical Zariski topology on Spec^{s}(M) for a right R-module M. We obtain a characterization of semisimple modules by using the second spectrum of a module. We prove that if R is a ring such that every right primitive factor of R is right artinian, then every non-zero submodule of a second right R-module M is second if and only if M is a fully prime module. We give some equivalent conditions for Spec^{s}(M) to be a Hausdorff space or T₁-space when the right R-module M has certain algebraic properties. We obtain characterizations of commutative Quasi-Frobenius and artinian rings by using topological properties of the second classical Zariski topology. We give a full characterization of the irreducible components of Spec^{s}(M) for a non-zero injective right module M over a ring R such that every prime factor of R is right or left Goldie.