On reversibility related to idempotents
Bull. Korean Math. Soc.
Published online 2019 Jan 14
Da Woon Jung, Chang Ik Lee, Yang Lee, Sangwon Park, Sung ju Ryu, Hyo Jin Sung, and Sang Jo Yun
Finance.Fishery.Manufacture Industrial Mathematics Center on Big Data, Institute of Basic Science, Daejin University, Department of Mathematics, Dong-A University, Department of Mathematics, Pusan National 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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