ABSTRACT
This chapter discusses ring theoretic finiteness conditions for semigroup algebra coming from certain descending chain conditions, including artinian, semiprimary, perfect, semilocal, local, and chain semigroup algebras. It presents a particular case of local semigroup algebras, that is, the algebras in which the Jacobson radical is a maximal ideal. The chapter gives a complete description of semigroup algebras that are perfect. An algebra R is right perfect if it is semilocal. Since there are known examples of semigroups that are right but not left nilpotent, there exist semigroup algebras that are right but not left perfect. The chapter also provides a description of the radical of finite dimensional semigroup algebras.
