The Zariski covering number for vector spaces and modules

Abstract

Given a module M over a commutative unital ring R, let denote the covering number, i.e. the smallest (cardinal) number of proper submodules whose union covers M; this includes the covering numbers of Abelian groups, which are extensively studied in the literature. Recently, Khare–Tikaradze [Comm. Algebra, in press] showed in several cases that where SM is the set of maximal ideals with Our first main result extends this equality to all R-modules with small Jacobson radical and finite dual Goldie dimension. We next introduce and study a topological counterpart for finitely generated R-modules M over rings R, whose ‘some’ residue fields are infinite, which we call the Zariski covering number To do so, we first define the “induced Zariski topology” τ on M, and now define to be the smallest (cardinal) number of proper τ-closed subsets of M whose union covers M. We then show our next main result: for all finitely generated R-modules M for which (a) the dual Goldie dimension is finite, and (b) whenever is finite. As a corollary, this alternately recovers the aforementioned formula for the covering number of the aforementioned finitely generated modules. Finally, we discuss the notion of κ-Baire spaces, and show that the inequalities again become equalities when the image of M under the continuous map (with appropriate Zariski-type topologies) is a κM -Baire subspace of the product space.