Endomorphism representations of Zorn rings and subclasses of rings with enough idempotents.
Let be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.
In this paper we investigate the related comparability over exchange rings. It is shown that an exchange ring R satisfies the related comparability if and only if for any regular x C R, there exists a related unit w C R and a group G in R such that wx C G.
We characterize exchange rings having stable range one. An exchange ring has stable range one if and only if for any regular , there exist an and a such that and if and only if for any regular , there exist and such that if and only if for any , .
It is shown that a ring is a -ring if and only if there exists a complete orthogonal set of idempotents such that all are -rings. We also investigate -rings for Morita contexts, module extensions and power series rings.
We proved in an earlier work that any existence variety of regular algebras is generated by its simple unital Artinian members, while any existence variety of Arguesian sectionally complemented lattices is generated by its simple members of finite length. A characterization of the class of simple unital Artinian members [members of finite length, respectively] of such varieties is given in the present paper.