Reversibility over upper nilradicals
Commun. Korean Math. Soc. 2020 Vol. 35, No. 2, 447-454
Published online April 6, 2020
Da Woon Jung, Chang Ik Lee, Zhelin Piao, Sung Ju Ryu, Hyo Jin Sung, Sang Jo Yun
Pusan National University; Pusan National University; Yanbian University; Pusan National University; Pusan National University; Dong-A University
Abstract : The studies of reversible and NI rings have done important roles in noncommutative ring theory. A ring $R$ shall be called {\it QRUR} if $ab = 0$ for $a,b \in R$ implies that $ba$ is contained in the upper nilradical of $R$, which is a generalization of the NI ring property. In this article we investigate the structure of QRUR rings and examine the QRUR property of several kinds of ring extensions including matrix rings and polynomial rings. We also show that if there exists a weakly semicommutative ring but not QRUR, then K\" othe's conjecture does not hold.
Keywords : QRUR ring, QRPR ring, upper nilradical, reversibile ring, K\"othe's conjecture, NI ring, weakly semicommutative ring
MSC numbers : 16N40, 16U80
Supported by : The first author was financially supported by NRF-2018R1D1A1B07048197.
The second author was financially supported by NRF-2019R1I1A3A01058630.
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