The Zariski covering number for vector spaces and modules
Article Type
Research Article
Publication Title
Communications in Algebra
Abstract
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.
First Page
1994
Last Page
2017
DOI
10.1080/00927872.2021.1995741
Publication Date
1-1-2022
Recommended Citation
Ghosh, Soham, "The Zariski covering number for vector spaces and modules" (2022). Journal Articles. 3404.
https://digitalcommons.isical.ac.in/journal-articles/3404
Comments
Open Access, Green