On strongly πregular rings and homomorphisms into them

Abstract

A ring S is strongly π-regular if for every a ∈ S there exists n ≥ 1 such that an∈ an+1S. It is first shown that the dominion and the maximal epimorphic extension of any ring homomorphism α:R →,S, S strongly π-regular, are strongly π-regular. Several results of Schofield on perfect and semiprimary rings are special cases. As an application it is shown that a strongly π-regular ring is a Schur ring, generalizing Lenagan’s theorem for artinian rings. Further if S ⊆ R where S is commutative strongly π-regular then for any finitely generated MSM⊗SR = 0, implies M = 0 , although ⊗Sis not faithful on homomorphisms. Another application shows that the dominion of the natural map Z → S acts like a characteristic for a strongly π-regular ring S A right noetherian ring R satisfying generic regularity, as defined by Goodearl, is shown to have a representation in a strongly π-regular ring [Rcirc] (which is also regular and biregular with simple images artinian) such that Spec [Rcirc] = Spec R , as sets, and Spec [Rcirc] ˜ Patch R , as topological spaces. Characterizations are given of when a commutative ring R embeds in a (commutative) strongly π-regular ring. Embeddability is equivalent to the existence of certain special, possibly infinite, primary decompositions of 0 in R . Several families of examples are given illustrating embeddability and non-embeddability.