ABSTRACT
A number of finiteness conditions have been investigated in the class of group and semigroup rings. None of those conditions implies the periodicity of the underlying semigroup. This chapter gives a brief survey of the main results concerning the most important finiteness conditions. If a unitary algebra R is a commutative domain, then, as is well known, R is hereditary if and only if R is a Dedekind domain, and R is semihereditary if and only if R is a Prüfer domain. The chapter formulates the result of semihereditariness of semigroup algebra only in the fundamental case of cancellative semigroups. A commutative semigroup S is a semilattice T of its archimedean components, which are cancellative if S is separative.
