On von Neumann Regularity of Commutators

Abstract

We study the structure of rings which satisfy the von Neumann regularity of commutators, and call a ring [Formula: see text] C-regular if [Formula: see text] for all [Formula: see text], [Formula: see text] in [Formula: see text]. For a C-regular ring [Formula: see text], we prove [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are the Jacobson radical, upper nilradical, Wedderburn radical, and center of a given ring [Formula: see text], respectively, and [Formula: see text] denotes the polynomial ring with a set [Formula: see text] of commuting indeterminates over [Formula: see text]; we also prove that [Formula: see text] is semiprime if and only if the right (left) singular ideal of [Formula: see text] is zero. We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular, from any given ring. Moreover, for a C-regular ring [Formula: see text], the following are proved to be equivalent: (i) [Formula: see text] is Abelian; (ii) every prime factor ring of [Formula: see text] is a duo domain; (iii) [Formula: see text] is quasi-duo; and (iv) [Formula: see text] is reduced.