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- Title
On von Neumann Regularity of Commutators.
- Authors
Kim, Nam Kyun; Kwak, Tai Keun; Lee, Yang; Ryu, Sung Ju
- Abstract
We study the structure of rings which satisfy the von Neumann regularity of commutators, and call a ring R C-regular if a b − b a ∈ (a b − b a) R (a b − b a) 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; (iii) R is quasi-duo; and (iv) R / W (R) is reduced.
- Subjects
VON Neumann, John, 1903-1957; COMMUTATION (Electricity); JACOBSON radical; PRIME factors (Mathematics); COMMUTATORS (Operator theory); POLYNOMIAL rings; COMMUTATIVE rings
- Publication
Algebra Colloquium, 2024, Vol 31, Issue 2, p181
- ISSN
1005-3867
- Publication type
Article
- DOI
10.1142/S1005386724000154