Regular Algebras
Jan Okniński · 2020
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 .