On von Neumann Regularity of Commutators

We study the structure of rings which satisfy the von Neumann regularity of commutators, and call a ring R C-regular if ab- ba is an element of (ab-ba)R(ab-ba) for all a, b in R. For a C-regular ring R, we prove J(R[X]) = N-& lowast; (R[X]) = N-& lowast; (R) [X] = W(R) [X] subset of Z(R[X]), where J(A), N-& lowast; (A), W(A), Z( A) are the Jacobson radical, upper nilradical, Wedderburn radical, and center of a given ring A, respectively, and A[X] denotes the polynomial ring with a set X of commuting indeterminates over A; we also prove that R is semiprime if and only if the right (left) singular ideal of R 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 R, the following are proved to be equivalent: (i) R is Abelian; (ii) every prime factor ring of R is a duo domain; (iii) R is quasi-duo; and (iv) R/ W( R) is reduced.