NJ-symmetric rings
Sanjiv Subba, Tikaram Subedi
TL;DR
The paper defines NJ-symmetric rings via $abc\in N(R)\Rightarrow bac\in J(R)$ for all $a,b,c\in R$ and explores their position relative to known ring classes. It develops a range of results showing when NJ-symmetry holds (e.g., for left/right quasi-duo, abelian $J$-clean, abelian $J$-quasipolar, and certain GWS rings) and how it behaves under constructions such as Morita contexts, Dorroh extensions, and matrix-extensions, while also proving that $M_n(R)$ is never NJ-symmetric for $n>1$ and that $R/J(R)$ NJ-symmetric implies $R$ is NJ-symmetric. The work also demonstrates that polynomial extensions can fail to preserve NJ-symmetry, providing explicit counterexamples, and examines how quotients, corners, and extensions interact with NJ-symmetry, including cases where the converse to certain implications does not hold. Overall, the results deepen understanding of how generalized symmetry properties interact with radical structures and standard ring-construction operations.
Abstract
We call a ring $R$ NJ-symmetric if $abc\in N(R)$ implies $bac\in J(R)$ for any $a,b,c\in R$. Some classes of rings that are NJ-symmetric include left (right) quasi-duo rings, weak symmetric rings, and abelian J-clean rings. We observe that if $R/J(R)$ is NJ-symmetric, then $R$ is NJ-symmetric, and therefore, we study some conditions for NJ-symmetric ring $R$ for which $R/J(R)$ is symmetric. It is observed that for any ring $R$, $M_n(R)$ is never an NJ-symmetric ring for all positive integer $n>1$. Therefore, matrix extensions over an NJ-symmetric ring is studied in this paper. Among other results, it is proved that there exists an NJ-symmetric ring whose polynomial extension is not NJ-symmetric.