MiS Preprint Repository

Considering finite extensions $K[A]\subseteq K[B]$ of positive affine semigroup rings over a field $K$ we have developed in [1] an algorithm to decompose $K[B]$ as a direct sum of monomial ideals in $K[A]$. By computing the regularity of homogeneous semigroup rings from the decomposition we have confirmed the Eisenbud-Goto conjecture in a range of new cases not tractable by standard methods. Here we first illustrate this technique and its implementation in our Macaulay2 package [MonomialAlgebras] by computing the decomposition and the regularity step by step for an explicit example. We then focus on ring-theoretic properties of simplicial semigroup rings. From the characterizations given in [1] we develop and prove explicit algorithms testing properties like Buchsbaum, Cohen-Macaulay, Gorenstein, normal, and seminormal, all of which imply the Eisenbud-Goto conjecture. All algorithms are implemented in our [Macaulay2] package.