The residue fields of a zero-dimensional ring Original Research Article

Gilmer and Heinzer have considered the question: For an indexed family of fields oK = {Kα}αgEA, under what conditions does there exist a zero-dimensional ring R (always commutative with unity) such that oK is up to isomorphism the family of residue fields {R/Mα}αgEA of R? If oK is the family of residue fields of a zero-dimensional ring R, then the associated bijection from the index set A to the spectrum of R (with the Zariski topology) gives A the topology of a Boolean space. The present paper considers the following question: Given a field F, a Boolean space X and a family {Kx}xgEX of extension fields of F, under what conditions does there exist a zero-dimensional F-algebra R such that oK is up to F-isomorphism the family of residue fields of R and the associated bijection from X to Spec(R) is a homeomorphism? A necessary condition is that given x in X and any finite extension E of F in Kx, there exist a neighborhood V of x and, for each y in V, an F-embedding of E into Ky. We prove several partial converses of this result, under hypotheses which allow the “straightening” of the F-embeddings to make them compatible. We give particular attention to the cases where X has only one accumulation point and where X is countable; and we provide several examples.