Rings in which the square of a unit is the sum of 1 and an element from $\sqrt{J(R)}$
Dinesh Udar, Shiksha Saini
TL;DR
The paper introduces the notion of $2-\\sqrt{J}U$ rings, rings where for every unit $u$, $u^2 \\in 1+\\sqrt{J(R)}$, and situates this class among known unit-related ring classes. It develops foundational properties of \\sqrt{J(R)}, demonstrates preservation under quotients, products, and corner/unit-closed subrings, and shows that $M_n(R)$ is never a $2-\\sqrt{J}U$ ring for $n>1$. It then analyzes matrix extensions (Morita contexts, triangular rings, and other constructions) and group rings, providing transfer principles and structural constraints, including that group rings of a $2-\\sqrt{J}U$ ring force the group to be a torsion, often a $2$-group, with consequences for the unit structure in $RG$. These results map the landscape of $2-\\sqrt{J}U$ rings and yield practical criteria for verifying the property in common algebraic constructions.
Abstract
Through this paper, we study the rings in which every unit's square is an element of the set $1+\sqrt{J(R)}$, and call them $2-\sqrt{J}U$ rings. Here, $\sqrt{J(R)}=\{x \in R: x^m \in J(R)$ for some $m \geq 1 \}$. We show that every $UU,~UJ,~2-UU,~2-UJ$ and $\sqrt{J}U$ ring is a $2-\sqrt{J}U$ ring. After exploring the basic properties, we show that the corner ring and unit closed subring of a $2-\sqrt{J}U$ ring are also $2-\sqrt{J}U$ rings. The ring of all $n\times n$ matrix rings for any $n>1$ is never a $2-\sqrt{J}U$ ring. We have focused on several other matrix extensions and group rings.