Abstract
The studies of reversible and $2$-primal rings have done important roles in noncommutative ring theory. We in this note introduce the concept of {\it quasi-reversible-over-prime-radical} (simply, {\it QRPR}) as a generalization of the $2$-primal ring property. A ring is called {\it QRPR} if $ab=0$ for $a, b\in R$ implies that $ab$ is contained in the prime radical. In this note we study the structure of QRPR rings and examine the QRPR property of several kinds of ring extensions which have roles in noncommutative ring theory.
Highlights
Throughout this note every ring is an associative ring with identity unless otherwise stated
Semicommutative rings are 2-primal through a simple computation, but the converse need not hold as can be seen by U2(D) over a 2-primal ring D, noting that U2(D) is 2-primal but non-Abelian
We will use Lemma 1.4 freely. 2-primal rings are QRPR by Lemma 1.1
Summary
Throughout this note every ring is an associative ring with identity unless otherwise stated. A ring R is called 2-primal if N∗(R) = N (R), following Birkenmeier, Heatherly, and E.K. Lee [6]. A prime ideal P of a ring R is called completely prime if R/P is a domain. A ring R is 2-primal if and only if every minimal prime ideal of R is completely prime, by [21, Proposition 1.11].
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