The Zariski covering number for vector spaces and modules

TL;DR

This paper extends the understanding of the minimal number of proper subspaces needed to cover vector spaces and modules, introducing a topological approach that generalizes existing algebraic results.

Contribution

It generalizes the formula for the covering number to modules with small Jacobson radical and finite dual Goldie dimension, and introduces a topological Zariski covering number for finitely generated modules.

Findings

The covering number formula applies to all modules with small Jacobson radical and finite dual Goldie dimension.

The Zariski covering number is bounded above by the algebraic covering number.

For finitely generated modules with certain properties, the Zariski and algebraic covering numbers coincide.

Abstract

Given a $K$-vector space $V$, let $\sigma(V,K)$ denote the covering number, i.e. the smallest (cardinal) number of proper subspaces whose union covers $V$. Analogously, define $\sigma(M,R)$ for a module $M$ over a unital commutative ring $R$; this includes the covering numbers of Abelian groups, which are extensively studied in the literature. Recently, Khare-Tikaradze [Comm. Algebra, in press] showed for several classes of rings $R$ and $R$-modules $M$ that $\sigma(M,R)=\min_{\mathfrak{m}\in S_M} |R/\mathfrak{m}| + 1$, where $S_M$ is the set of maximal ideals $\mathfrak{m}$ such that $\dim_{R/\mathfrak{m}}(M/\mathfrak{m}M)\geq 2$. (That $\sigma(M,R)\leq\min_{\mathfrak{m}\in S_M}|R/\mathfrak{m}|+1$ is straightforward.) Our first main result extends this equality to all $R$-modules with small Jacobson radical and finite dual Goldie dimension. We next introduce 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 $\sigma_\tau(M,R)$. To do so, we first define the "induced Zariski topology" $\tau$ on $M$, and now define $\sigma_\tau(M,R)$ to be the smallest (cardinal) number of proper $\tau$-closed subsets of $M$ whose union covers $M$. We first show that our choice of topology implies that $\sigma_\tau(M,R)\leq\sigma(M,R)$, the covering number. We then show our next main result: $\sigma_\tau(M,R)=\min_{\mathfrak{m}\in S_M} |R/\mathfrak{m}|+1$, for all finitely generated $R$-modules $M$ for which (a) the dual Goldie dimension is finite, and (b) $\mathfrak{m}\notin S_M$ whenever $R/\mathfrak{m}$ is finite. As a corollary, this alternately recovers the above formula for the covering number $\sigma(M,R)$ of the aforementioned finitely generated modules. We also extend these topological studies to general finitely generated $R$-modules, using the notion of $\kappa$-Baire spaces.