On right regularity of commutators
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We study the structure of right regular commutators, and call a ring $R$ {\it strongly $C$-regular} if $ab-ba\in (ab-ba)^2R$ for any $a, b\in R$. We first prove that a noncommutative strongly $C$-regular domain is a division algebra generated by all commutators; and that a ring (possibly without identity) is strongly $C$-regular if and only if it is Abelian $C$-regular (from which we infer that strong $C$-regularity is left-right symmetric). It is proved that for a strongly $C$-regular ring $R$, (i) if $R/W(R)$ is commutative, then $R$ is commutative; and (ii) every prime factor ring of $R$ is either a commutative domain or a noncommutative division ring, where $W(R)$ is the Wedderburn radical of $R$.
Abstract
We study the structure of right regular commutators, and call a ring $R$ {\it strongly $C$-regular} if $ab-ba\in (ab-ba)^2R$ for any $a, b\in R$. We first prove that a noncommutative strongly $C$-regular domain is a division algebra generated by all commutators; and that a ring (possibly without identity) is strongly $C$-regular if and only if it is Abelian $C$-regular (from which we infer that strong $C$-regularity is left-right symmetric). It is proved that for a strongly $C$-regular ring $R$, (i) if $R/W(R)$ is commutative, then $R$ is commutative; and (ii) every prime factor ring of $R$ is either a commutative domain or a noncommutative division ring, where $W(R)$ is the Wedderburn radical of $R$.
참고문헌 (19)
- G. Azumaya, "Strongly π-regular rings", J. Fac. Sci. Hokkaido Univ. Ser. I, 13, pp.34~39 (1954)
- H. E. Bell, "Near-rings in which each element is a power of itself", Bull. Austral. Math. Soc, 2, pp.363~368 (1970)[DOI]
- A. W. Chatters, "Rings with chain conditions, Research Notes in Mathematics" (1980), Pitman
- 최광진, "Reversibility and symmetry over centers", 대한수학회지, 56(3), pp.723~738 (2019), 대한수학회[DOI]
- P. M. Cohn, "Reversible rings", Bull. London Math. Soc, 31(6), pp.641~648 (1999)[DOI]
- J. L. Dorroh, "Concerning adjunctions to algebras", Bull. Amer. Math. Soc, 38(2), pp.85~88 (1932)[DOI]
- S. Fukuda, "On rings in which all commutators are strongly regular", Math. J. Okayama Univ, 27, pp.39~44 (1985)
- K. R. Goodearl, "von Neumann regular rings, Monographs and Studies in Mathematics" (1979), Pitman
- K. R. Goodearl, "An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts" (1989), Cambridge University Press
- I. N. Herstein, "A condition for the commutativity of rings", Canadian J. Math, 9, pp.583~586 (1957)[DOI]
- C. Huh, "Basic examples and extensions of symmetric rings", J. Pure Appl. Algebra, 202(1-3), pp.154~167 (2005)[DOI]
- C. Huh, "p. p. rings and generalized p. p. rings", J. Pure Appl. Algebra, 167(1), pp.37~52 (2002)[DOI]
- N. K. Kim, "On von Neumann regularity of commutators", ~
- N. K. Kim, "Extensions of reversible rings", J. Pure Appl. Algebra, 185(1-3), pp.207~223 (2003)[DOI]
- T. Y. Lam, "A First Course in Noncommutative Rings", Graduate Texts in Mathematics (1991), Springer US[DOI]
- J. Lambek, "On the representation of modules by sheaves of factor modules", Canad. Math. Bull, 14, pp.359~368 (1971)[DOI]
- G. Marks, "Reversible and symmetric rings", J. Pure Appl. Algebra, 174(3), pp.311~318 (2002)[DOI]
- J. V. Neumann, "On regular rings", Proceedings of the National Academy of Sciences, 22, pp.707~713 (1936)
- M. S. Putcha, "Structure of rings satisfying certain identities on commutators", Proc. Amer. Math. Soc, 32, pp.57~62 (1972)[DOI]