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.