The Zariski covering number for vector spaces and modules

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.