Rings in which the square of a unit is the sum of 1 and an element from $\sqrt{J(R)}$

TL;DR

This paper introduces and studies a new class of rings called $2-\sqrt{J}U$ rings, characterized by units whose squares are in $1+\sqrt{J(R)}$, exploring their properties and substructures.

Contribution

The paper defines $2-\sqrt{J}U$ rings, establishes their basic properties, and shows how certain subrings and extensions relate to this class, expanding the understanding of ring structures.

Findings

Every $UU,~UJ,~2-UU,~2-UJ$ and $\sqrt{J}U$ ring is a $2-\sqrt{J}U$ ring.

Corner rings and unit-closed subrings of a $2-\sqrt{J}U$ ring are also $2-\sqrt{J}U$ rings.

The ring of all $n\times n$ matrices for $n>1$ is never a $2-\sqrt{J}U$ ring.

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.