2-UQ Rings and Unit Square Structures

• 2-UQ rings are unital rings defined by every unit’s square equaling 1 plus a quasi‑nilpotent element that commutes with it.
• They generalize 2‑UJ and 2‑UU rings while interlinking key invariants such as the Jacobson radical, nilpotency, and tripotency in R/J(R).
• Their structure is preserved under constructions like subrings, corners, and trivial extensions, but fails in nontrivial matrix rings.

A 2‑UQ ring is a unital ring in which the square of every unit can be written as the sum of the identity and a quasi‑nilpotent element that commutes with it. This property generalizes the structure observed in unit groups for several other classes of “clean-like” and unipotent-like rings, and interacts deeply with classical invariants such as the Jacobson radical, nilpotency, and the presence of certain identities (e.g., tripotency). The property is robust under a wide range of ring-theoretic constructions and provides a unifying viewpoint for extensions of nil-clean and unipotent unit rings.

1. Definition and Basic Properties

A ring $R$ (associative with unity) is called a 2‑UQ ring if every unit $u \in U(R)$ satisfies

$u^2 = 1 + q$

for some $q \in QN(R)$, with $QN(R)$ the set of quasi-nilpotent elements of $R$. Equivalently, for all $u \in U(R)$, $(u^2 - 1)$ is quasi-nilpotent and commutes with $u^2$. Quasi-nilpotent elements are those $q \in R$ such that for every $x \in R$ commuting with $q$, $1 - qx$ is invertible.

2‑UQ rings properly include:

• 2‑UJ rings ($u^2 = 1 + j$, $j \in J(R)$)
• 2‑UU rings ($u^2 = 1 + n$, $n \in Nil(R)$)
• UQ rings ($u = 1 + q$, $q \in QN(R)$)

Key structural facts include:

• In potent rings, the 2‑UQ, 2‑UU, and 2‑UJ properties coincide, all characterized by tripotency of $R/J(R)$.
• The class is strictly larger than 2‑UU or 2‑UJ rings, as there exist rings which are 2‑UQ but not 2‑UU or 2‑UJ (Najafi et al., 14 Sep 2025, Mahmood et al., 8 Aug 2025).

2. Relationship with Related Ring Classes

The 2‑UQ property sits naturally in a lattice of "unit square" conditions. The inclusions and distinctions are summarized as follows:

Ring Class	Unit Square	Constraints on $u^2$
2‑UU	$u^2$	$u^2 = 1 + n$, $n \in Nil(R)$
2‑UJ	$u^2$	$u^2 = 1 + j$, $j \in J(R)$
2‑UQ	$u^2$	$u^2 = 1 + q$, $q \in QN(R)$
2‑UNJ (see (Mahmood et al., 8 Aug 2025))	$u^2$	$u^2 = 1 + n + j$, $n \in Nil(R), j \in J(R)$
UQ	$u$	$u = 1 + q$, $q \in QN(R)$

The following implications hold:

• $2$‑$\text{UU} \implies 2$‑$\text{UQ}$
• $2$‑$\text{UJ} \implies 2$‑$\text{UQ}$
• The converse implications do not hold in general; explicit examples are given in (Najafi et al., 14 Sep 2025, Mahmood et al., 8 Aug 2025).

In potent or semi‑potent rings, the chain collapses: for $R$ potent, $R$ is 2‑UQ iff $R/J(R)$ is tripotent, and this is also equivalent to $R$ being 2‑UJ or 2‑UU (Najafi et al., 14 Sep 2025).

3. Structural Results and Preservation under Constructions

The 2‑UQ property is preserved under numerous constructions:

• Subrings: If $S \subseteq R$ is a good subring (i.e., $U(R) \cap S \subseteq U(S)$) of a 2‑UQ ring, $S$ is also 2‑UQ (Najafi et al., 14 Sep 2025).
• Corners: If $R$ is 2‑UQ and $e^2 = e \in R$, then $eRe$ is 2‑UQ (Najafi et al., 14 Sep 2025).
• Direct Products: $R = \prod_{i} R_i$ is 2‑UQ iff every $R_i$ is 2‑UQ.
• Trivial and Triangular Extensions: Trivial extensions $T(R, M)$ and triangular matrix rings $T_n(R)$ are 2‑UQ iff $R$ is 2‑UQ.
• Morita Contexts: If $R$ is a Morita context with connecting bimodules $M, N$ such that $MN$ and $NM$ are nilpotent and central, $R$ is 2‑UQ iff both end rings are 2‑UQ (Najafi et al., 14 Sep 2025).
• Matrix rings: For $n \geq 2$, $M_n(S)$ is never 2‑UQ if $S \ne 0$ (Najafi et al., 14 Sep 2025); more generally, any ring with a non-trivial matrix corner (of size at least $2$) cannot be 2‑UQ.

This non-preservation for matrix rings is parallel to results on UQ rings (Danchev et al., 23 Feb 2024), where higher-dimensional block structures obstruct quasi-nilpotent square decompositions.

4. Interactions with Regularity, Potence, and Cleanness

The 2‑UQ condition is intricately interconnected with standard ring-theoretic regularity properties:

• Potent and Semi‑Potent Rings: For a potent or semi‑potent ring $R$,
  • $R$ is 2‑UQ $\iff R/J(R)$ is tripotent ($a^3 = a$ for all $a \in R/J(R)$);
  • In such cases, $R$ is also 2‑UJ and 2‑UU (Najafi et al., 14 Sep 2025).
• Regular Rings: In a regular 2‑UQ ring, equivalences $R$ regular $\iff$ $R$ strongly regular $\iff$ $R$ unit-regular $\iff$ $R$ tripotent$$hold.</li> <li><strong>Clean/Exchange Properties</strong>: Many 2‑UQ rings are clean, typically when$$R/J(R)$$is tripotent; strongly clean, clean, and quasi-nil-clean decompositions become equivalent under the identification$$QN(R) = J(R)$$, which holds for semipotent rings (<a href="/papers/2402.15455" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Danchev et al., 23 Feb 2024</a>, <a href="/papers/2509.11319" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Najafi et al., 14 Sep 2025</a>).</li> </ul> <h2 class='paper-heading' id='2-uq-group-rings-and-their-restrictions'>5. 2‑UQ Group Rings and Their Restrictions</h2> <p>A key focus in recent research is the behavior of the 2‑UQ property in group rings$$RG$:</p> <ul> <li>If$RG$is 2‑UQ and$2 \in J(R)$, then$G$$must be a 2‑group (<a href="/papers/2509.11319" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Najafi et al., 14 Sep 2025</a>).</li> <li>If$$3 \in J(R)$, then$G$must be a 3‑group or have all elements of order$2$(exponent$2$); that is,$G$$is either a 3-group or an elementary abelian 2-group.</li> <li>For$$R = \mathbb{Z}_m$,$R$is 2‑UQ if and only if$m = 2^k 3^s$for some$k,s \ge 0$$(<a href="/papers/2509.11319" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Najafi et al., 14 Sep 2025</a>).</li> <li>These constraints mirror those for strongly nil-clean and n‑UU group rings (<a href="/papers/2311.15018" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Danchev et al., 2023</a>), further delineating the effect of the Jacobson radical primes on admissible underlying groups.</li> </ul> <p>The necessity arises from the forced structure on units and the requirement that$$u^2 - 1$always lands in$QN(RG)$$, strongly impacting possible group exponents.</p> <h2 class='paper-heading' id='boundary-examples-extensions-and-generalizations'>6. Boundary Examples, Extensions, and Generalizations</h2> <p>Multiple explicit constructions demonstrate the strict enlargement of the 2‑UQ class over its relatives:</p> <ul> <li>In$$A = \mathbb{F}_3\langle x, y \mid x^2 = 0 \rangle$, every unit squared lands in$1 + QN(A)$, so$A$is 2‑UQ but not 2‑UJ because$J(A) = 0$and, for instance,$(1+x)^2 \notin 1 + J(A)$$(<a href="/papers/2509.11319" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Najafi et al., 14 Sep 2025</a>).</li> <li>$$B = \mathbb{F}_2[[x]]$is 2‑UQ,$1 + x$is a unit,$(1 + x)^2 = 1 + x^2$, but$B$is not 2‑UU since$x^2$$is not nilpotent (<a href="/papers/2509.11319" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Najafi et al., 14 Sep 2025</a>).</li> <li>Matrix rings:$$M_{n}(S)$with$n \geq 2$$always fail to be 2‑UQ (<a href="/papers/2402.15455" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Danchev et al., 23 Feb 2024</a>, <a href="/papers/2509.11319" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Najafi et al., 14 Sep 2025</a>), reflecting the essential one-dimensionality of 2‑UQ behavior.</li> </ul> <p>The class 2‑UNJ (units square to$$1 + n + j$with$n$nilpotent,$j \in J(R)$) and 2‑ΔU (unit squares in$1 + A(R)$, with$A(R)$$the largest Jacobson radical subring stable under multiplication by units) generalize or straddle the 2‑UQ property, and current research is investigating the precise relationships and hierarchies between these classes (<a href="/papers/2508.06689" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Mahmood et al., 8 Aug 2025</a>, <a href="/papers/2501.04720" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Hasanzadeh et al., 2 Jan 2025</a>).</p> <h2 class='paper-heading' id='connections-to-quadratic-algebras-and-involutive-structures'>7. Connections to Quadratic Algebras and Involutive Structures</h2> <p>Beyond unit-square phenomena, the concept of a universal quadratic structure associated to a standard involution on an algebra is intimately connected to the 2‑UQ pattern in noncommutative settings (<a href="/papers/1003.3512" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Voight, 2010</a>). For an$$R$-algebra$B$$of degree 2, one always has a canonical involutive antiautomorphism$$\overline{\phantom{x}}$such that every$x \in B$satisfies</p> <p>$x^2 - (x + \overline{x})x + x\overline{x} = 0.$$</p> <p>This universal quadratic equation, governed by involution, characterizes—under mild hypotheses—algebras of degree 2 and is central to the classification of quadratic and exceptional rank‑3 algebras.</p> <p><em>This suggests that in the broadest sense, the unifying thread of 2‑UQ behavior is the universal constraint of a quadratic identity linking units (or elements) to distinguished subsets such as$$QN(R)$$or the image of a canonical involution.</em></p> <hr> <p><strong>Summary Table: Preservation and Non-Preservation of the 2‑UQ Property</strong></p> <div class='overflow-x-auto max-w-full my-4'><table class='table border-collapse w-full' style='table-layout: fixed'><thead><tr> <th>Construction</th> <th>2‑UQ Preservation</th> </tr> </thead><tbody><tr> <td>Direct product$$\prod_{i} R_i$</td> <td>Yes iff each$R_i$ 2‑UQ Trivial extension $T(R, M)$</td> <td>Yes iff$R$ 2‑UQ Triangular matrix ring $T_n(R)$</td> <td>Yes iff$R$ 2‑UQ Matrix ring $M_n(R)$,$n \geq 2$ Never 2‑UQ Morita context with nilpotent trace Yes iff both corners 2‑UQ

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The theory of 2‑UQ rings now underpins a host of connections between unit behavior, radical theory, and polynomial and matrix extension phenomena, with numerous research directions in the classification of potential counter-examples, criteria for group rings, and the systematic paper of regular, potent, and clean ring contexts (Najafi et al., 14 Sep 2025, Danchev et al., 23 Feb 2024, Mahmood et al., 8 Aug 2025, Hasanzadeh et al., 2 Jan 2025, Voight, 2010).