向量空间与模的Zapriki覆盖数
The Zariski covering number for vector spaces and modules
摘要：
给定一个$K$ -向量空间$V$，令$σ(V,K)$表示覆盖数，即其并集覆盖$V$的真子空间的最小（基数）数。类似地，定义$σ(M,R)$为有单位交换环$R$上的模$M$;这包括阿贝尔群的覆盖数，这在文献中得到了广泛的研究。最近，Khare-Tikaradze [Comm.代数，出版中]表明，对于几类环$R$和$R$ -模块$M$，$σ(M,R)=\min_{\mathfrak{m}\in S_M} |R/\mathfrak{m}| + 1$，其中$S_M$是极大理想$\mathfrak{m}$的集合，使得$\dim_{R/\mathfrak{m}}(M/\mathfrak{m}M)\geq 2$。(That $σ(M,R)\leq\min_{\mathfrak{m}\in S_M}|R/\mathfrak{m}|+1$很简单。）我们的第一个主要结果将这个等式推广到所有具有小Jacobson根和有限对偶Goldie维数的$R$ -模。接下来我们引入环$R$上的生成的$R$ -模$M$的一个拓扑对应物，它的“一些”剩余域是无限的，我们称之为Zerkiki覆盖数$σ_τ(M,R)$。为此，我们首先在$M$上定义“诱导Zeroki拓扑”$τ$，现在定义$σ_τ(M,R)$为$M$的真$τ$闭子集的最小（基数）数，其并集覆盖$M$。我们首先表明，我们的拓扑结构的选择意味着$σ_τ(M,R)\leqσ(M,R)$，覆盖数。然后，我们显示我们的下一个主要结果：$σ_τ(M,R)=\min_{\mathfrak{m}\in S_M} |R/\mathfrak{m}|+1$，对于所有生成的$R$ -模块$M$，其中（a）对偶Goldie维数是有限的，和（b）$\mathfrak{m}\notin S_M$每当$R/\mathfrak{m}$是有限的。作为推论，这交替地恢复了用于上述随机生成的模块的覆盖数$σ(M,R)$的上述公式。我们也将这些拓扑研究推广到一般的$R$ -模，使用$κ$-Baire空间的概念。
英文摘要：
Given a $K$-vector space $V$, let $σ(V,K)$ denote the covering number, i.e. the smallest (cardinal) number of proper subspaces whose union covers $V$. Analogously, define $σ(M,R)$ 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 $σ(M,R)=\min_{\mathfrak{m}\in S_M} |R/\mathfrak{m}| + 1$, where $S_M$ is the set of maximal ideals $\mathfrak{m}$ such that $\dim_{R/\mathfrak{m}}(M/\mathfrak{m}M)\geq 2$. (That $σ(M,R)\leq\min_{\mathfrak{m}\in S_M}|R/\mathfrak{m}|+1$ 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 $σ_τ(M,R)$. To do so, we first define the "induced Zariski topology" $τ$ on $M$, and now define $σ_τ(M,R)$ to be the smallest (cardinal) number of proper $τ$-closed subsets of $M$ whose union covers $M$. We first show that our choice of topology implies that $σ_τ(M,R)\leqσ(M,R)$, the covering number. We then show our next main result: $σ_τ(M,R)=\min_{\mathfrak{m}\in S_M} |R/\mathfrak{m}|+1$, 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 $σ(M,R)$ of the aforementioned finitely generated modules. We also extend these topological studies to general finitely generated $R$-modules, using the notion of $κ$-Baire spaces.
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