On reversibility related to idempotents
Bull. Korean Math. Soc. 2019 Vol. 56, No. 4, 993-1006
Published online 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
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