Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2020; 35(2): 447-454

Online first article January 20, 2020      Printed April 30, 2020

https://doi.org/10.4134/CKMS.c190109

Copyright © The Korean Mathematical Society.

Reversibility over upper nilradicals

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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