ABSTRACT
This chapter examines self-injective semigroup algebras. A ring R is right self-injective if R is an injective right R-module. This is an important property, strongly related to von Neumann regularity, and coinciding with the quasi-Frobenius property when restricted to finite dimensional unitary algebras. The chapter provides examples of finite commutative monoids with non-self-injective semigroup algebras. It presents theorems that provide solutions to the main problem for (right and left) self-injective semigroup algebras. The chapter discusses the characterization of Frobenius algebras to provide a criterion for determining Frobenius semigroup algebras. It also exploits the fact that finite dimensional unitary right self-injective algebras are quasi-Frobenius.
