ABSTRACT
This chapter discusses the dimensions of semigroup algebras. It also discusses the classical Krull dimension, the Gelfand-Kirillov dimension, and the Krull dimension for the semigroup algebras. While the general problem of computing a given dimension in terms of the underlying semigroup has not been settled for group algebras, the chapter is restricted to the class of semigroup algebras satisfying polynomial identities. As usual, when considering the Krull dimensions, the chapter deals with algebras with unity. Therefore, for the sake of simplicity, monoid algebras are mainly discussed. The description of the classical Krull dimension for semigroup algebras of commutative cancellative semigroups is generalized to arbitrary commutative semigroups.
