ABSTRACT
In this chapter, we discuss the von Neumann regularity condition for the class of semigroup algebras. Recall that an algebra R is called regular if, for every a ∈ R, there exists x ∈ R such that axa = a. While every semisimple artinian algebra is regular, the results of this chapter may be viewed as generalizations of the description of semisimple artinian semigroup algebras given in Chapter 14. For the general theory of regular algebras, we refer to [69]. We note that R is regular if and only if every finitely generated right ideal of R is of the form eR for some e = e 2 ∈ R. In particular, R satisfies the right (and similarly left) semihereditariness condition, which is reviewed in Chapter 17.
