Hereditary semigroup rings and maximal orders

Abstract

ln this thesis, we study several problems concerning semigroup algebras K[S] of a semigroup S over a field K. -- In Chapter 1 and Chapter 2 we give some background on semigroups and semigroup rings. In Chapter 3, we discuss the global dimension of semigroup rings R[S] where R is a ring and S is a monoid with a sequence of ideals S = I₁ ⊃ I₂ ⊃ ⋅⋅⋅ ⊃ In ⊃ In₊₁ such that each Ii/Ii₊₁ is a non-null Rees matrix semigroup. -- In Chapter 4, we investigate when a semigroup algebra has right global dimension at most 1, that is, when is it right hereditary. As an application of the results in Chapter 3, we describe when K[S¹] is hereditary for a non-null Rees semigroup S. For arbitrary semigroups that are nilpotent in the sense of Malcev, we describe when its semigroup algebra is hereditary Noetherian prime. And for cancellative semigroups we obtain a description of when its semigroup algebra is hereditary Noetherian. -- In Chapter 5, we generalize the concept of unique factorization monoid and investigate Noetherian unique factorization semigroup algebras of submonoids of torsion-free polycyclic-by-finite groups. -- In Chapter 6, we investigate when a semigroup algebra K[S] is a polynomial identity domain which is also a unique factorization ring. In order to prove this result we describe first the height one prime ideals of such algebras.