DECOMPOSING SEMIGROUP ALGEBRAS

Abstract

Suppose that A and B are cancellative abelian semigroups and that R is an integral domain.We demonstrate that this
semigroupring R[B] may be reduced to a direct sum of R[A]-submodules of the ring of R[Aquotient] as a R[A]-module. In
the case of a finite extension of positive affine semigroup rings, we provide a method for calculating the decomposition.This
decomposition allows us to calculate different ring-theoretic features of R[B] for polynomial rings over fields, and we
demonstrate how to do so for R[A]. Specifically, we provide a fast method for determining the Castelnuovo-Mumford
regularity of homogeneous semigroup rings.In a number of novel contexts, we find evidence that supports the Eisenbud-
Goto theory. Our algorithms are part of the MACAULAY2 package MONOMIALALGEBRAS.