On von Neumann Regularity of Commutators

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.