On strongly #x3C0;regular rings and homomorphisms into them

摘要

A ring S is strongly #x3C0;-regular if for every a #x2208; S there exists n #x2265; 1 such that an#x2208; an+1S. It is first shown that the dominion and the maximal epimorphic extension of any ring homomorphism #x3B1;:R #x2192;,S, S strongly #x3C0;-regular, are strongly #x3C0;-regular. Several results of Schofield on perfect and semiprimary rings are special cases. As an application it is shown that a strongly #x3C0;-regular ring is a Schur ring, generalizing Lenagan#x2019;s theorem for artinian rings. Further if S #x2286; R where S is commutative strongly #x3C0;-regular then for any finitely generated MSM#x2297;SR = 0, implies M = 0 , although #x2297;Sis not faithful on homomorphisms. Another application shows that the dominion of the natural map Z #x2192; S acts like a characteristic for a strongly #x3C0;-regular ring S