Journal of the
Korean Mathematical Society JKMS

We introduce two subclasses of abelian McCoy rings, so-called $\pi$-\textit{CN}-rings and $\pi$-duo rings, and systematically study their fundamental characteristic properties accomplished with relationships among certain classical sorts of rings such as $2$-primal rings, bounded rings etc. It is shown that a ring $R$ is $\pi$-\textit{CN} whenever every nilpotent element of index $2$ in $R$ is central. These rings naturally generalize the long-known class of \textit{CN}-rings, introduced by Drazin \cite{drz}. It is proved that $\pi$-\textit{CN}-rings are abelian, McCoy and $2$-primal. We also show that, $\pi$-duo rings are strongly McCoy and abelian and also they are strongly right $AB$. If $R$ is $\pi$-duo, then $R[x]$ has property ($A$). If $R$ is $\pi$-duo and it is either right weakly continuous or every prime ideal of $R$ is maximal, then $R$ has property ($A$). A $\pi$-duo ring $R$ is left perfect if and only if $R$ contains no infinite set of orthogonal idempotents and every left $R$-module has a maximal submodule. Our achieved results substantially improve many existing results.

Keywords: $CN$-rings, $\pi$-$CN$-rings, 2-primal rings, McCoy rings, duo-rings, $\pi$-duo rings, strongly $AB$-rings, property ($A$), nil radical

MSC numbers: 16S34, 16U99, 16E50, 16W10, 13B99