The Zariski covering number for vector spaces and modules

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 $sigma(M,R)leqmin_{mathfrak{m}in S_M}|R/mathfrak{m}|+1$ 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 $sigma_tau(M,R)leqsigma(M,R)$, 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) $mathfrak{m} otin S_M$ 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.