Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Subba, Sanjiv, Subedi, Tikaram
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4407570027