On von Neumann Regularity of Commutators  

作　　者：Nam Kyun Kim Tai Keun Kwak Yang Lee Sung Ju Ryu Nanqing Ding 

机构地区：[1]School of Basic Sciences,Hanbat National University,Daejeon 34158,Korea [2]Department of Data Science,Daejin University,Pocheon 11159,Korea [3]Department of Mathematics,Yanbian University,Yanji,Jilin 133002,China [4]Institute for Applied Mathematics and Optics,Hanbat National University Daejeon 34158,Korea [5]Department of Mathematics,Pusan National University,Busan 46241,Korea [6]不详

出　　处：《Algebra Colloquium》2024年第2期181-198,共18页代数集刊（英文版）

摘　　要：We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R)[X]=W(R)[X]■Z(R[X]),where J(A),N^(*)(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;(ii)R is quasi-duo;and(iv)R/W(R)is reduced.

关 键 词：C-regular ring COMMUTATOR regular ring commutative ring RADICAL singular ideal 

分 类 号：O17[理学—数学]