Prüfer rings

Abstract

In the introduction to his book, Multiplicative Ideal Theory [26], Robert Gilmer states: "It is possible to enumerate a few concepts which are central in our development of multiplicative ideal theory. Quotient rings and rings of quotients fall into this category, they are basic to all subsequent considerations; invertible ideals also constitute a basic tool in the presentation of the theory. A third concept which plays a central role in the development of the classical ideal theory is that of a Priifer domain". Priifer domains were defined in 1932 by H. Priifer [56], as domains in which every finitely generated ideal is invertible. In 1936, Krull [49] named these rings in Prufer's honor and proved the first of the many equivalent conditions that make an integral domain Priifer (see Theorem 1.1). Although, the new concept started to slowly appear in the literature, it reached its central role which it enjoys today in the sixties and seventies, due, in no small part, to Robert Gilmer's publications and their impact on research in commutative algebra. On one hand, Priifer rings and related ring conditions feature in many of Robert's about 200 articles, investigating a large variety of ring properties, including the connection of the Priifer condition to other properties of interest, see for example [5, 6, 18-36]. Many commutative algebraists followed in Robert's footsteps, using some of his methods and examples, including his emphasis on the Priifer domain notion. On the other hand, in 1968, the first version of Robert's book: Multiplicative Ideal Theory [26], was published by Queen's University Press, collecting all results and references known to date on Priifer rings and emphasizing their central role in ring theory research. This was followed by the publication of revised versions of this book in 1972 [29], and in 1992 [36]. This book became, and continues to be, one of the most influential books for research in commutative algebra, and with it the centrality of the Priifer domain notion becomes consolidated.