Commutative rngs

Abstract

The purpose of this article is to discuss commutative rngs (that is, commutative rings that do not have an identity) and especiaUy Robert Gilmer's work in this area. To avoid confusion we adopt the following terminology. The word "ring" will be used in the neutral sense that there may or may not be an identity. The term "rng" or "ring without identity" will mean a ring that does not have an identity and we will always explicitly say "ring with identity" when that is the case. The term "rng" appears in Jacobson [37, Section 2.17] where he suggests the pronunciation riing and says that the term was suggested to him by Louis Rowen. Bourbaki [6, Chapter 1] uses the term "pseudo-ring"for rings without identity. While we will be mostly concerned with commutative rings, in several places we consider noncommutative rings. This should be clear from context. Today the word "ring" usually means ring with identity. This was not the case so long ago. I remember reading Lambek's book Lectures on Rings and Modules [38] in 1968 and being taken back by the fact that he assumed the existence of an identity element. When I taught material on the Jacobson radical thirty years ago I did it via quasi-regular elements. Today I usually assume the existence of an identity. I suspect the insistence on an identity today is in part due to the larger role played by homological methods in ring theory. Also, the existence of an identity entails the quasi-compactness of the spectrum of a ring, a useful property for algebraic geometry. Now there are plenty of good rings that don't have an identity such as the even integers 2Z and an infinite direct sum Rα of rings. On the other hand, since any ring can naturally be embedded in a ring with identity (see Section 2), many mathematicians no doubt take the point of view that there is no loss in generality in assuming the existence of an identity. The extent to which this is true depends on the context. When asked to write an article for this volume, I gave considerable thought to the choice of topics. Certainly a number of suitable topics came to mind: dimension theory, Priifer domains and valuation domains, polynomial rings and power series rings, and semigroup rings. But I finally chose commutative rings without identity. About thirty of Robert's almost two hundred papers involve rngs to some extent or another - twenty-six which involve rngs in a significant way are listed in the references. One can of course debate which papers to include since many results on rings do not depend on the existence of an identity. Perhaps I picked commutative rngs for a topic because I remembered the material on rngs in Chapter 1 of Multiplicative Ideal Theory [25] and consulting his paper "Eleven nonequivalent conditions on a commutative ring" [15] (maybe even chuckling at the title).