Local rings with zero-dimensional formal fibers

Subrings of zero-dimensional rings

When Sarah Glaz, Bill Heinzer and the junior author of this article approached Robert with the idea of editing a book dedicated to his work, we asked him to give us a list of his work and to comment on it to the extent he felt comfortable. As usual, he was extremely thorough in his response. When the authors of this article began to consider what topic we wanted to write about, we were impresse...

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-dimens...

Infinite Products of Zero-dimensional Commutative Rings

Zero-dimensional Subrings of Commutative Rings

We consider various conditions of the sets of zero-dimensional, Artinian, and von Neumann regular subrings of a commutative ring. Section 1 treats questions of existence of such rings, Section 2 deals with the situation in which all subrings belong to one of the three classes, and Section 3 is concerned with the behavior of the sets under intersection. In Section 4 we give a brief survey of som...

COMPLETIONS OF UFDs WITH SEMI-LOCAL FORMAL FIBERS

Let T M be a complete local ring such that T/M = T . Given a finite set of incomparable nonmaximal prime ideals C of T , we provide necessary and sufficient conditions for T to be the completion of a local UFD A, whose generic formal fiber is semilocal with maximal ideals the elements of C. We also show that, given the T above, we can find necessary and sufficient conditions for T to be the com...