The Zariski covering number for vector spaces and modules

Article Type

Research Article

Publication Title

Communications in Algebra


Given a module M over a commutative unital ring R, let (Formula presented.) 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 (Formula presented.) where SM is the set of maximal ideals (Formula presented.) with (Formula presented.) 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 (Formula presented.) To do so, we first define the “induced Zariski topology” τ on M, and now define (Formula presented.) to be the smallest (cardinal) number of proper τ-closed subsets of M whose union covers M. We then show our next main result: (Formula presented.) for all finitely generated R-modules M for which (a) the dual Goldie dimension is finite, and (b) (Formula presented.) whenever (Formula presented.) is finite. As a corollary, this alternately recovers the aforementioned formula for the covering number (Formula presented.) of the aforementioned finitely generated modules. Finally, we discuss the notion of κ-Baire spaces, and show that the inequalities (Formula presented.) again become equalities when the image of M under the continuous map (Formula presented.) (with appropriate Zariski-type topologies) is a κM -Baire subspace of the product space.

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Open Access, Green

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