The family of residue fields of a zero-dimensional commutative ring
Let \({\mathcal F}\) denote a family of fields. The theme of this paper is the realizability of \({\mathcal F}\): that is, under what conditions on \({\mathcal F}\) does there exist a commutative zero-dimensional ring \(R\) such that the family \({\mathcal F}(R)\) of residue fields of \(R\) is in bijection with \({\mathcal F}\)? For example, it is shown that if \({\mathcal F}\) includes a finite subset \(\{F_ 1,\ldots,F_ n\}\) such that every field in \({\mathcal F}\) contains an isomorphic copy of (at least) one of the \(F_ i\), then \({\mathcal F}\) is realizable. In the reverse direction, if \({\mathcal F}\) consists of finite fields, or if every field in \({\mathcal F}\) is a finite dimensional vector space over a fixed field \(K\), then \({\mathcal F}\) is realizable only if it includes a finite set of minimal members. However, this converse result fails in general: For every prime \(p\) there is a realizable set \({\mathcal F}=\{K_ i:0 \leq i<\infty\}\) with each \(K_ i\) of characteristic \(p\), \(K_ 0\) infinite, and all \(K_ i\) distinct, finite and minimal, for \(i \geq 1\).
family of residue fields of a commutative zero-dimensional ring
scientific article; zbMATH DE number 166140
scientific article