Rings with 2-$\Delta$U property

TL;DR

This paper investigates a special class of rings called 2-ΔU rings, characterized by the property that the square of each unit lies in a specific radical-related subset, exploring their structure and behavior under algebraic constructions.

Contribution

It introduces and studies the properties and structure of 2-ΔU rings, expanding understanding of their relation to other ring classes and their behavior under algebraic operations.

Findings

2-ΔU rings include several known classes like UJ-rings and ΔU-rings.

ΔU-rings are shown to be UUC.

The structure of 2-ΔU rings is characterized under various conditions.

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.