Residue Fields of Zero-Dimensional Rings

Robert Gilmer

Given a zero-dimensional commutative ring $R$, we investigate the structure of the family $\calf(R)$ of residue fields of $R$. We show that if a family $\calf$ of fields contains a finite subset $\{F_1,\ldots,F_n)$ such that every field in $\calf$ contains an isomorphic copy of at least one of the fields $F_i$, then there exists a zero-dimensional reduced ring $R$ such that $\calf=\calf(R)$. If every residue field of $R$ is a finite field, or is a finite-dimensional vector space over a fixed field $K$, we prove, conversely, that the family $\calf(R)$ has, to within isomorphism, finitely many minimal elements.