On reversibility related to idempotents
Bull. Korean Math. Soc. 2019 Vol. 56, No. 4, 993-1006
Published online July 9, 2019
Printed July 31, 2019
Da Woon Jung, Chang Ik Lee, Yang Lee, Sangwon Park, Sung Ju Ryu, Hyo Jin Sung, Sang Jo Yun
Pusan National University; Pusan National University; Daejin University; Dong-A University; Pusan National University; Pusan National University; Dong-A University
Abstract : This article concerns a ring property which preserves the reversibility of elements at nonzero idempotents. A ring $R$ shall be said to be {\it 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.
Keywords : quasi-reversible ring, Abelian ring, reversible ring, matrix ring, polynomial ring, NI ring
MSC numbers : 16U80, 16S36, 16S50
Supported by : The first named author was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korean Government (MSIP)(NRF-2017R1A5A1015722). The third named author was supported by the National Natural Science Foundation of China(11361063)
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