IDEMPOTENTS IN π-REGULAR RINGS, RIGHT AI RINGS, NI RINGS AND GENERALIZED REGULAR RINGS

Von Neumann regular rings are studied by ring theorists and functional analysts in connection with operator algebra theory. In particular, the concept of idempotent in algebra is a generalization of projection in analysis. We study the structure of idempotents in 7r-regular rings, right AI rings (i.e., for every element a, ab is an idempotent for some nonzero element b), NI rings, and generalized regular rings (i.e., every nonzero principal right ideal contains a nonzero idempotent). We obtain a well-known fact, proved by Menal, Nicholson and Zhou, that idempotents can be lifted modulo every ideal in 7r-regular rings, as a corollary of one of main results of this article. It is shown that the 7r-regularity is seated between right AI and regularity. We also show that from given any 7r-regular ring, we can construct a right AI ring but not 7r-regular. In addition, we study the structure of idempotents of 7r-regular rings and right AI rings in relation to the ring properties of Abelian and NI, giving simpler proofs to well-known results for Abelian 7r-regular rings.