Noetherianity of the space of irreducible representations

Abstract

Let R be an associative ring with identity. We study an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple left R-modules (or, more generally, simple objects in a complete abelian category). Under this topology the points are closed, and when R is left noetherian the corresponding topological space is noetherian. If R is commutative (or PI, or FBN) the corresponding topological space is naturally homeomorphic to the maximal spectrum, equipped with the Zariski topology. When R is the first Weyl algebra (in characteristic zero) we obtain a one-dimensional irreducible noetherian topological space. Comparisons with topologies induced from those on A. L. Rosenberg's spectra are briefly noted.