Rings with 2-$\Delta$U property

Abstract

Rings in which the square of each unit lies in $1+\Delta (R)$, are said to be $2$-$\Delta U$, where $J(R)\subseteq \Delta (R)=:\{r\in R|r+U(R)\subseteq U(R)\}$. The set $\Delta (R)$ is the largest Jacobson radical subring of $R$ which is closed with respect to multiplication by units of $R$ and is studied in \cite{2}. The class of $2$-$\Delta U$ rings consists several rings including $UJ$-rings, $2$-$UJ$ rings and $\Delta U$-rings, and we observe that $\Delta U$-rings are $UUC$. The structure of $2$-$\Delta U$ rings is studied under various conditions. Moreover, the $2$-$\Delta U$ property is studied under some algebraic constructions.

18 pages. arXiv admin note: substantial text overlap with arXiv:2411.09416