Symmetricity and reversibility from the perspective of nilpotents
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In this paper, we deal with the question that what kind of properties does a ring gain when it satisfies symmetricity or reversibility by the way of nilpotent elements? By the motivation of this question, we approach to symmetric and reversible property of rings via nilpotents. For symmetricity, we call a ring $R$ {\it middle right-}(resp.~{\it left-}){\it nil symmetric} (mr-nil (resp.~ml-nil) symmetric, for short) if $abc = 0$ implies $acb = 0$ (resp.~$bac = 0)$ for $a$, $c\in R$ and $b\in $ nil$(R)$ where nil$(R)$ is the set of all nilpotent elements of $R$. It is proved that mr-nil symmetric rings are abelian and so directly finite. We show that the class of mr-nil symmetric rings strictly lies between the classes of symmetric rings and weak right nil-symmetric rings. For reversibility, we introduce {\it left} (resp.~{\it right}) {\it N-reversible ideal} $I$ of a ring $R$ if for any $a\in$ nil$(R)$, $b\in R$, being $ab \in I$ implies $ba \in I$ (resp.~$b\in$ nil$(R)$, $a\in R$, being $ab \in I$ implies $ba \in I$). A ring $R$ is called {\it left} (resp.~{\it right}) {\it N-reversible} if the zero ideal is left (resp.~right) N-reversible. Left N-reversibility is a generalization of mr-nil symmetricity. We exactly determine the place of the class of left N-reversible rings which is placed between the classes of reversible rings and CNZ rings. We also obtain that every left N-reversible ring is nil-Armendariz. It is observed that the polynomial ring over a left N-reversible Armendariz ring is also left N-reversible.
Abstract
In this paper, we deal with the question that what kind of properties does a ring gain when it satisfies symmetricity or reversibility by the way of nilpotent elements? By the motivation of this question, we approach to symmetric and reversible property of rings via nilpotents. For symmetricity, we call a ring $R$ {\it middle right-}(resp.~{\it left-}){\it nil symmetric} (mr-nil (resp.~ml-nil) symmetric, for short) if $abc = 0$ implies $acb = 0$ (resp.~$bac = 0)$ for $a$, $c\in R$ and $b\in $ nil$(R)$ where nil$(R)$ is the set of all nilpotent elements of $R$. It is proved that mr-nil symmetric rings are abelian and so directly finite. We show that the class of mr-nil symmetric rings strictly lies between the classes of symmetric rings and weak right nil-symmetric rings. For reversibility, we introduce {\it left} (resp.~{\it right}) {\it N-reversible ideal} $I$ of a ring $R$ if for any $a\in$ nil$(R)$, $b\in R$, being $ab \in I$ implies $ba \in I$ (resp.~$b\in$ nil$(R)$, $a\in R$, being $ab \in I$ implies $ba \in I$). A ring $R$ is called {\it left} (resp.~{\it right}) {\it N-reversible} if the zero ideal is left (resp.~right) N-reversible. Left N-reversibility is a generalization of mr-nil symmetricity. We exactly determine the place of the class of left N-reversible rings which is placed between the classes of reversible rings and CNZ rings. We also obtain that every left N-reversible ring is nil-Armendariz. It is observed that the polynomial ring over a left N-reversible Armendariz ring is also left N-reversible.
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