Symmetricity and reversibility from the perspective of nilpotents

Commun. Korean Math. Soc. Published online November 9, 2020

Abdullah Harmanci, Handan Kose, and Burcu Ungor
Hacettepe University, Kirsehir Ahi Evran University, Ankara University

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.

Keywords : Symmetric ring; middle right-nil symmetric ring; nil-symmetric ring; reversible ring; left N-reversible ring