Some research on chains of prime ideals influenced by the writings of Robert Gilmer

Abstract

Robert Gilmer's work has overtly influenced the direction of much of commutative ring theory for the past 40-plus years. Although I have not worked on many of the explicit questions that Robert raised, I feel that Robert's work has definitely infiuenced my research, both in identifying important topics and themes and in exemplifying the highest standards of exposition. I hope that the next two sections serve to document that influence and to indicate where it has led me and others. Each of the following sections begins with a theorem of Robert Gilmer and then proceeds to examine a cross-section of subsequent developments. The first of these sections concerns various characterizations of Priifer domains, beginning with a result of Robert for the one-dimensional case, and features a discussion of going-down domains, treed domains and universally catenarian domains. The final section concerns more recent work. It begins by explaining how an exquisitely finitistic result that Robert obtained jointly with Nashier-Nichols spurred my interest in infinite chains and trees of prime ideals of arbitrary (commutative) rings. The section goes on to discuss several directions of inquiry, including an application to treed domains. We use the following notation and conventions. R denotes a (commutative integral) domain with integral closure R' and quotient field K. "Dimension" refers to Krull dimension and is denoted by dimv valuative dimension is denoted by dim^. All rings are assumed commutative with identity. Notation such as A ⊆ B means that A is a (unital) subring of B, while the notation f:A → B means that is a (unital) ring homomorphism from A to B. If A is a ring, then by an averring of A, we mean an (unital) A-subalgebra of the total quotient ring of A. Modifying the usage in [52, page 28], we use GD, GU, LO and INC to refer to the going-down, going-up, lying-over and incomparable properties, respectively, of ring extensions or ring homomorphisms. As usual, the cardinality of a set S is denoted by |S|.