Semigroup algebras and direct sums of domains

Abstract

Let k be a field and S a commutative semigroup. Assume the semigroup algebra k[S] has an identity. We characterize those semigroup algebras which are direct sums of domains. As applications, we deduce necessary and sufficient conditions for a torsion-free semigroup algebra to a Krull order, a Gaussian ring, or a hereditary Noetherian ring.