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ON REVERSIBILITY RELATED TO IDEMPOTENTS

  • Jung, Da Woon (Finance Fishery Manufacture Industrial Mathematics Center on Big Data Pusan National University) ;
  • Lee, Chang Ik (Department of Mathematics Pusan National University) ;
  • Lee, Yang (Department of Mathematics Yanbian University) ;
  • Park, Sangwon (Department of Mathematics Dong-A University) ;
  • Ryu, Sung Ju (Department of Mathematics Pusan National University) ;
  • Sung, Hyo Jin (Department of Mathematics Pusan National University) ;
  • Yun, Sang Jo (Department of Mathematics Dong-A University)
  • Received : 2018.08.14
  • Accepted : 2018.11.21
  • Published : 2019.07.31

Abstract

This article concerns a ring property which preserves the reversibility of elements at nonzero idempotents. A ring R shall be said to be quasi-reversible if $0{\neq}ab{\in}I(R)$ for a, $b{\in}R$ implies $ba{\in}I(R)$, where I(R) is the set of all idempotents in R. We investigate the quasi-reversibility of 2 by 2 full and upper triangular matrix rings over various kinds of reversible rings, concluding that the quasi-reversibility is a proper generalization of the reversibility. It is shown that the quasi-reversibility does not pass to polynomial rings. The structure of Abelian rings is also observed in relation with reversibility and quasi-reversibility.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF), National Natural Science Foundation of China

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