Residue Fields of Zero-Dimensional Rings

We begin by deening a couple of terms. If F = fF i g i2I is a family of elds, we say that F is the family of residue elds of the ring T if there exists a bijection g : I ! MaxSpec(T) such that F i ' T=g(i) for each i 2 I. Note that the deenition takes into account the multiplicity with which a given eld occurs in F. We say that F is realizable if F is the family of residue elds of a zero-dimensional ring. This paper is concerned with the following question (RF). (RF) What families of elds are realizable? At the outset we can say that while much is known in regard to (RF), the general case of the question remains open, and at this point there is no conjectured answer to (RF). Questions of this type have not received much attention in the study of commutative rings, and it seems worthwhile to ask how (RF) arose. For Heinzer and me, there were two primary sources of motivation. The rst of these was Theorem 12 of I], which states that a ring S is hereditarily zero-dimensional This paper contains material presented in the third 1994 Barrett Lecture, given on April 9. Most of its results come from the paper GH6] (see the list of references for the paper Background and Preliminaries : : : in this volume), and represent work done jointly with W. Heinzer.