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Rings Such That $u-1$ Lies In $J^{\#}(R)$ For Each Unit $u$

Peter Danchev, Mina Doostalizadeh, Mehrdad Esfandiar, Omid Hasanzadeh

TL;DR

This work introduces $UJ^{\#}$ rings, where each unit can be written as $u=1+j$ with $j\in J^{\#}(R)$, and studies their place among UU and UJ rings. It provides extensive characterizations, showing that $R$ is $UJ^{\#}$ iff $R/J(R)$ is UU and exploring the Boolean quotient case under semi-potent assumptions, along with stability under common constructions and consequences for ring-theoretic properties such as Dedekind-finiteness and cleanness. The paper also derives strong restrictions for group rings, proving that $RG$ being $UJ^{\#}$ forces $G$ to be a $2$-group (with equivalences in the locally finite case) and analyzes how augmentation and radicals behave in these contexts. Collectively, the results illuminate how unit structure interacts with radical-like parts of a ring and constrain composite constructions like matrix and group rings, offering a unified framework around $UJ^{\#}$ rings.

Abstract

We investigate the so-called {\it $UJ^{\#}$ rings}, a new type of rings in which every unit can be written as $1+j$ with $j\in J^{\#}(R)$. These rings were defined and studied by Saini-Udar in Czechoslovak Math. J. (2025) under the name {\it $\sqrt{J}U$ rings}. (See \cite{SU}.) This class extends both the classes of UU and UJ rings, but also has its own special properties. In this study, we present some additional results about $UJ^{\#}$ rings that supply those from \cite{SU} explaining their connections with Dedekind-finite, semi-potent and Boolean rings, respectively, as well as we give several characterizations in this direction. We also examine how these rings behave under common ring constructions and find conditions for group rings to be $UJ^{\#}$. Moreover, our establishments shed a clearer picture of how unit elements interact with radical-like parts of a ring.

Rings Such That $u-1$ Lies In $J^{\#}(R)$ For Each Unit $u$

TL;DR

This work introduces rings, where each unit can be written as with , and studies their place among UU and UJ rings. It provides extensive characterizations, showing that is iff is UU and exploring the Boolean quotient case under semi-potent assumptions, along with stability under common constructions and consequences for ring-theoretic properties such as Dedekind-finiteness and cleanness. The paper also derives strong restrictions for group rings, proving that being forces to be a -group (with equivalences in the locally finite case) and analyzes how augmentation and radicals behave in these contexts. Collectively, the results illuminate how unit structure interacts with radical-like parts of a ring and constrain composite constructions like matrix and group rings, offering a unified framework around rings.

Abstract

We investigate the so-called {\it rings}, a new type of rings in which every unit can be written as with . These rings were defined and studied by Saini-Udar in Czechoslovak Math. J. (2025) under the name {\it rings}. (See \cite{SU}.) This class extends both the classes of UU and UJ rings, but also has its own special properties. In this study, we present some additional results about rings that supply those from \cite{SU} explaining their connections with Dedekind-finite, semi-potent and Boolean rings, respectively, as well as we give several characterizations in this direction. We also examine how these rings behave under common ring constructions and find conditions for group rings to be . Moreover, our establishments shed a clearer picture of how unit elements interact with radical-like parts of a ring.
Paper Structure (4 sections, 51 theorems, 30 equations)

This paper contains 4 sections, 51 theorems, 30 equations.

Table of Contents

  1. Introduction and Motivation
  2. Examples and Fundamental Properties of $UJ^{\#}$ Rings
  3. Main Results
  4. Group $UJ^{\#}$ Rings

Key Result

Lemma 2.1

Let $R$ be any ring. Then, the following assertions hold: (1) If $a \in J^{\#}(R)$ and $b \in R$ with $ab = ba$, then $ab \in J^{\#}(R)$. (2) For every $a \in R$ and $n \in \mathbb{N}$, we have $a^n \in J^{\#}(R)$ if and only if $a \in J^{\#}(R)$. (3) If $a \in J^{\#}(R)$, then $1 - a \in U(R)$. (4)

Theorems & Definitions (95)

  • Lemma 2.1
  • proof
  • Example 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • ...and 85 more