Reversibility over upper nilradicals
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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.
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.
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