Numerical semigroup algebras

Abstract

Given a ring R and a semigroup S the semigroup ring R[S] inherits the properties of S and R. If no restrictions are posed on the semigroup S and on the ring R the class of semigroup rings is very large. There is a wide literature on this subject. Gilmer's book [15] is the classical reference in the commutative case, i.e. when both the ring R and the semigroup S are commutative. The book contains a deep study of the conditions under which the semigroup ring R[S] has given ring theoretic properties. The following short survey considers a very particular class of semigroup rings: the ring of coefficient R is supposed to be a field and, for the main part of the results, the semigroup S is supposed to be a numerical semigroup, i.e. an additive submonoid of N, with finite complement in N. Also within this particular class of semigroup algebras over a field, the paper deals only with some themes and it is far from being complete. Results and proofs are ranged through two different sources. On one side there is the classical ring theory, a rich, historically settled and well known theory. On the other side there are more elementary but simetimes quicker arguments which come from studying a less rich structure as that of semigroups. In doing so I hope to be not too far from R. Gilmer's open research attitude and from his tolerant view. Here, as in many other fields of human life, there is not a "Unique Thought", but several different points of view and techniques may coexist and be reciprocally useful.