CLASSICAL ZARISKI TOPOLOGY OF MODULES AND SPECTRAL SPACES I

Abstract

Let R be a ring, M be a left R-module and Spec(RM) be the collection of all prime submodules of M. In this paper and its sequel, we introduce and study a generalization of the Zariski topology of rings to modules and call it classical Zariski topology of M. Then we investigate the interplay between the module-theoretic properties of M and the topological properties of Spec(RM). Modules whose classical Zariski topology is respectively T1, Hausdorff or cofinite are studied, and several characterizations of such modules are given. We investigate this topological space from the point of view of spectral spaces (that is, topological spaces homeomorphic to the prime spectrum of a commutative ring equipped with the Zariski topology). We show that Spec(RM) is always a T0-space and each finite irreducible closed subset of Spec(RM) has a generic point. Then by applying Hochster’s characterization of a spectral space, we show that for each left R-module M with finite spectrum, Spec(RM) is a spectral space. In Part II we shall continue the study of this construction.

Keywords

• Prime submodule, Prime spectrum, Classical Zariski topology, Spectral space