Some abelian McCoy rings
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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.
Abstract
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.
참고문헌 (33)
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