Bulletin of the
Korean Mathematical Society BKMS

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 $\pi$-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 $\pi$-regular rings, as a corollary of one of main results of this article. It is shown that the $\pi$-regularity is seated between right AI and regularity. We also show that from given any $\pi$-regular ring, we can construct a right AI ring but not $\pi$-regular. In addition, we study the structure of idempotents of $\pi$-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 $\pi$-regular rings.

Keywords: Right AI ring, $\pi$-regular ring, idempotent, NI ring, idempotent-lifting, Abelian ring, generalized regular ring, matrix ring

MSC numbers: 16D25, 16S50, 16U70

Supported by: This work was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) grant funded by the Ministry of Education(NRF-2021R1I1A3045371). This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2022R1A5A1033624).