Robert Gilmer's work on semigroup rings

Abstract

Group rings, and more generally semigroup rings, have played an important role in modern algebra and topology. In this article, we are interested in Robert Gilmer's pioneering work on semigroup rings. This includes his two papers with T. Parker [30, 31] on divisibility properties in semigroup rings, submitted in March and May of 1973, respectively; his semigroup ring example of a two-dimensional non-Noetherian UFD [24], submitted in May of 1973; his work with J. T. Arnold on the (KruU) dimension of semigroup rings [12], submitted in September of 1975; and his book Commutative Semigroup Rings [25], finished in the summer of 1983 and published in 1984. Arnold and Parker (see [45]) were both PhD students of Gilmer. In the introduction, we give a leisurely motivation for semigroup rings and establish notation. In the second section, we cover Gilmer's work with T. Parker on divisibility properties in semigroup rings. In the third section, we discuss Gilmer's construction of a two-dimensional non-Noetherian UFD, his work with J. T. Arnold on the dimension of a semigroup ring, and his book on semigroup rings. In the final section, we consider generalizations to KruU semigroup rings, graded rings, and divisibility properties in semigroups. We also discuss the (t-)class group and Picard group of monoid domains. The polynomial ring Q[X] over the field Q of rational numbers is a PID. Varying the coefficients produces different ring-theoretic properties. For example, the polynomial ring D[X] over an integral domain D is a UFD (resp., GCD-domain, KruU domain, PVMD) if and only if > is a UFD (resp., GCDdomain, KruU domain, PVMD). One often tries to prove that -D[X] satisfies a certain property P if and only if D satisfies property P. Sometimes this holds; other times it does not. For example, because of dimension constraints, D[X] is a PID or a Dedekind domain only in the trivial case when D is a field.