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arXiv · 2602.14521

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

Dinesh Udar·Shiksha Saini
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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.

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Dinesh Udar, Shiksha Saini. 2026-02-16. Rings in which the square of a unit is the sum of 1 and an element from $\sqrt{J(R)}$. https://arxiv.org/abs/2602.14521

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