On Von Neumann Regular Rings And Jacobson’s Lemma

This paper gives characterizations of regular rings, unit-regular rings and strongly regular rings from the viewpoint of endomorphisms of modules. Relations between IC rings and regular rings, strongly regular rings and abelian regular rings, strongly regular elements and group invertible elements are also presented. Based on these results, the paper concludes the researches on the generalizations of Jacobson’s lemma for inner inverses, group inverses, Drazin inverses and generalized Drazin inverses. Some interesting results about spectral idempotents, associated idempotents and Jacobson pairs are given.