Abstract:

Rings in which the square of each unit lies in $1+\Delta(R)$ are said to be {\it $2$-$\Delta U$ rings}, 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 detailed studied in \cite{2}. The class of $2$-$\Delta U$ rings consists several rings including $UJ$-rings, $2$-$UJ$ rings and $\Delta U$-rings, respectively, and we observe that $\Delta U$-rings are $UUC$ in terms of \cite{13}. Furthermore, the structure of $2$-$\Delta U$ rings is examined under various algebraic conditions. Moreover, the $2$-$\Delta U$ property is explored under some extended constructions.

The established by us achievements substantially improved on the existing in the literature relevant results.