A STUDY ON TRI REVERSIBLE RINGS

Articles in Press

Document Type : Original Manuscript

Authors

Department of Mathematics, Gauhati University, Guwahati-781014, India.

Abstract

This article embodies a ring theoretic property which, preserves the reversibility of elements at non-zero tripotents. A ring R is defined as quasi tri reversible if any non-zero tripotent element ab of R implies ba is also a tripotent element in R for a, b ∈ R. We explore the quasi tri reversibility of 2 by 2 full and upper triangular matrix rings over various kinds of reversible rings, deducing that the quasi tri reversibility is a proper generalization of reversible rings. It is proved that the polynomial rings are not quasi tri reversible rings. The relation of symmetric rings, IF P and Abelian rings with reversibility and quasi tri reversibility arestudied. It is also observed that the structure of weakly tri normal rings and quasi tri reversible rings are independent of each other.

Keywords

References

  1. D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra, 27(6) (1999), 2847–2852.
  1. H. E. Bell, Near-rings in which each element is a power of itself, Bull. Aust. Math. Soc., 2 (1970), 363–368.
  1. P. M. Cohn, Reversible rings, Bull. London Math. Soc., 31(6) (1999), 641–648.
  2. K. R. Goodearl, Von Neumann Regular Rings, Monographs and Studies in Mathematics, 4, Pitman (Advanced Publishing Program), Boston, MA, 1979.
  1. H. M. I. Hoque and H. K. Saikia, A study on weakly tri normal and quasi tri normal rings, Palest. J. Math., 12(2) (2023), 125–132.
  1. C. Huh, H. K. Kim and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra, 167(1) (2002), 37–52.
  1. S. U. Hwang, Y. C. Jeon and Y. Lee, Structure and topological conditions of NI rings, J. Algebra, 302(1) (2006), 186–199.
  1. D. W. Jung, N. K. Kim, Y. Lee and S. J. Ryu, On properties related to reversible rings, Bull. Korean Math. Soc., 52(1) (2015), 247–261.
  1. D. W. Jung, C. I. Lee, Y. Lee, S. Park, S. J. Ryu, H. J. Sung and S. J. Yun, On reversibility related to idempotents, Bull. Korean Math. Soc., 56(4) (2019), 993–1006.
  1. N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra, 223 (2000), 477–488.
  1. N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra, 185 (2003) 207–223.
  1. T. K. Kwak, S. I. Lee and Y. Lee, Quasi-normality of idempotents on nilpotents, Hacet. J. Math. Stat., 48(6) (2019), 1744–1760.
  2. J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull., 14 (1971), 359–368.
  1. G. Marks, On 2-primal Ore extensions, Comm. Algebra, 29 (2001), 2113–2123.
  2. G. Marks, Reversible and symmetric rings, J. Pure Appl. Algebra, 174(3) (2002), 311–
  3. G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc., 184 (1973), 43–60.
  1. A. Smoktunowicz, Polynomial rings over nil rings need not be nil, J. Algebra, 233(2) (2000), 427–436.
  2. J. Von Neumann, On regular rings, Proceedings of the National Academy of Sciences, 22 (1936), 427-436
Journal of Algebraic Systems

Articles in Press, Accepted Manuscript
Available Online from 09 April 2024

Files

Share

How to cite

Statistics