Generalizations of semiregular rings

摘要

In this article, we call a ring R right generalized semiregular if for any, a E R there exist two left ideals P, L of R such that lr(a) = P circle plus L, where P subset of Ra and Ra boolean AND L is small in R. The class of generalized semiregular rings contains an semiregular rings and all AP-injective rings. Some properties of these rings are studied and some results about semiregular rings and AP-injective rings are extended. In addition, we call a ring R semi-pi-regular if for any, a E R there exist a positive integer n and e(2) = e is an element of a(n)R such that (1 - e)a(n) is an element of J(R), the Jacobson radical of R. It is shown that a ring R is semi-pi-regular if and only if R/J(R) is pi-regular and idempotents can he lifted modulo J(R).