Some New Classes of Rings Which Have the McCoy Condition
Peter Danchev, M. Zahiri
TL;DR
This paper generalizes reversibility through the notion of weakly reversible rings, showing that every weakly reversible ring is abelian and $2$-primal, hence McCoy, and that $Nil_{*}(R)=Nil(R)$ with $Nil(R[x])=Nil(R)[x]$. It provides a concrete example of a weakly reversible ring that is not reversible, illustrating the nontrivial gap between the two concepts. A key technical achievement is proving that $R[x]$ is strongly AB when $R$ is weakly reversible, which implies McCoy-ness and yields zip-transfer results between $R$ and $R[x]$, including the preservation of AB and zip properties under certain hypotheses. The results extend the understanding of zero-divisors, annihilators, and radical structure in noncommutative rings and their polynomial extensions, offering new tools for investigating McCoy-type behavior in broader classes of rings.
Abstract
We define here the notion of a {\it weakly reversible ring} $R$ saying that a non-zero element $a\in R$ is weakly reversible if there exists an integer $m>0$ depending on $a$ such that $a^m\neq 0$ is reversible, that is, $r_R(a^m)=l_R(a^m)$. In addition, $R$ is weakly reversible if all its elements are weakly reversible. It is shown that all weakly reversible rings are abelian McCoy rings and so, particularly, they are abelian 2-primal rings. Moreover, we construct a weakly reversible ring which is {\it not} reversible. We also show that if $R$ is a weakly reversible ring, then the polynomial ring $R[x]$ is strongly AB. Thus, in particular, the weakly reversible ring $R$ is zip if, and only if, $R[x]$ is zip. We, moreover, prove that if $R$ is a weakly reversible ring and every prime ideal of $R$ is maximal, then both $R$ and $R[x]$ are AB rings.