The residue fields of a zero-dimensional ring
Let \({\mathcal K} = \{K_{\alpha}\}_{\alpha \in A}\) denote an indexed family of fields. In J. Pure Appl. Algebra 82, No. 2, 131-153 (1992; Zbl 0782.13014), \textit{R. Gilmer} and \textit{G. Heinzer} considered the question under what conditions there exists a zero-dimensional commutative ring \(R\) such that \(\mathcal K\) is the family of residue fields \(\{R/M_{\alpha}\}_{\alpha \in A}\) up to isomorphism? In the case \(\mathcal K\) is the family of the residue fields of a zero-dimensional commutative ring \(R\) the associated bijection from the index set \(A\) to the spectrum of \(R\) (endowed with the Zariski topology) gives \(A\) the topology of a Boolean space. The paper under review is concerned with the following problem: Given a field \(F,\) a Boolean space \(X\) and a family \(\{K_x\}_{x \in X}\) of extension fields of \(F\). Under which circumstances does there exist a zero-dimensional \(F\)-algebra \(R\) such that \(\mathcal K\) is up to \(F\)-isomorphism the family of residue fields of \(R\) such that the associated bijection from \(X\) to \(\text{Spec} (R)\) is a homomorphism? The authors consider the necessary condition: For a given \(x \in X\) and any finite field extension \(F \subseteq E \subseteq K_x,\) there exists a neighborhood \(V\) of \(x\) and, for each \(y \in V,\) an \(F\)-embedding of \(E\) into \(K_x.\) Then they prove several partial converses of this result, under a hypothesis which allows the ``straightening'' of the \(F\)-embeddings to make them compatible. The authors give particular attention to the case where \(X\) has only one accumulation point and where \(X\) is countable. Moreover, several examples are provided.
scientific article; zbMATH DE number 1308189