Rings with 2-$Δ$U property
Omid Hasanzadeh, Ahmad Moussavi, Peter Danchev
TL;DR
This work introduces and develops the 2-$ΔU$ class of rings, extending the ΔU framework by requiring that the square of every unit is congruent to 1 modulo the Δ-subring, i.e., $u^2=1+r$ with $r\in Δ(R)$. It establishes foundational links between ΔU and traditional unit-structure concepts (notably $J(R)$, $Δ(R)$, and Boolean quotients $R/J(R)$), and derives a range of characterizations and equivalences under various hypotheses (potent, Artinian, finite, local). The paper then undertakes a detailed study of 2-$ΔU$ rings under classical constructions, proving stability (or lack thereof) through extensions such as skew polynomial/power series rings, Morita contexts, triangular and trivial extensions, generalized matrix rings, and group rings, with precise conditions for preserving the property. It provides concrete results showing that 2-$ΔU$ is often controlled by the quotient $R/J(R)$ (e.g., local rings with $R/J(R)\cong \mathbb{Z}_2$ or $\mathbb{Z}_3$) and supplies several structural classifications for semisimple and Artinian cases. The work closes with open questions aimed at refining the theory, including $2$-$UQ$ and $2$-$UNJ$ variants and their regularity classifications, inviting further exploration of unit-square behavior in this broader radical-closure context.
Abstract
Rings in which the square of each unit lies in $1+Δ(R)$, are said to be $2$-$ΔU$, where $J(R)\subseteqΔ(R) =: \{r \in R | r + U(R) \subseteq U(R)\}$. The set $Δ(R)$ 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 $2$-$ΔU$ rings consists several rings including $UJ$-rings, $2$-$UJ$ rings and $ΔU$-rings, and we observe that $ΔU$-rings are $UUC$. The structure of $2$-$ΔU$ rings is studied under various conditions. Moreover, the $2$-$ΔU$ property is studied under some algebraic constructions.