Potent 2-UQ Rings: Structure & Implications

Updated 18 September 2025

Potent 2-UQ rings are rings where every unit u satisfies u² = 1 + q with q quasi-nilpotent, featuring idempotent lifting and tripotency in R/J(R).
They generalize 2-UJ and 2-UU rings, linking regularity, cleanness, and semi-tripotency through a unified structural framework.
These rings preserve their properties under common constructions and connect to perfect number theory in quadratic rings, offering rich applications in modern algebra.

A potent 2-UQ ring is a ring $R$ in which every unit $u$ satisfies $u^2 = 1 + q$ for some quasi-nilpotent element $q \in QN(R)$ , and the ring is potent, meaning that idempotents lift modulo the Jacobson radical and every one-sided ideal not contained in $J(R)$ contains a nonzero idempotent. This property connects the unit-square condition to structural properties traditionally associated with regular, semi-tripotent, and clean rings. The investigation of potent 2-UQ rings formalizes and extends phenomena observed in both perfect number theory (in the context of quadratic rings) and ring-theoretic studies regarding the interplay between units, radicals, and nilpotence.

1. Definition and Formalism

A 2-UQ ring is defined as follows: for every $u \in U(R)$ , there exists $q \in QN(R)$ such that

$u^2 = 1 + q,$

where $QN(R)$ denotes the set of quasi-nilpotent elements (i.e., elements $x$ for which $1 - rx$ is invertible for all $r \in R$ ). In the potent context, $R$ is further assumed to satisfy conditions such as lifting idempotents modulo $J(R)$ and the existence of sufficiently many idempotents. The central structural theorem in this context is that for potent rings,

$\text{$R$ is 2-UQ} \iff R/J(R) \text{ is tripotent} \ (\forall a, \, a^3 = a \in R/J(R)),$

and, consequently, potent 2-UQ rings are precisely the semi-tripotent rings (Najafi et al., 14 Sep 2025).

The class of 2-UQ rings generalizes several previously studied classes:

2-UJ rings: $u^2 = 1 + j$ with $j \in J(R)$ . Since $J(R) \subseteq QN(R)$ , every 2-UJ ring is 2-UQ.
2-UU rings: $u^2 = 1 + n$ with $n$ nilpotent. Since $Nil(R) \subseteq QN(R)$ , every 2-UU ring is 2-UQ.
UNJ rings: $U(R) = 1 + Nil(R) + J(R)$ , broader than UU or UJ (Mahmood et al., 8 Aug 2025). However, the converse inclusions do not hold in general, as shown through specific counterexamples. For instance, there exist 2-UQ rings that are neither 2-UJ nor 2-UU.

Table: Comparison of Unit Conditions

Ring Class	Unit Condition	Inclusion Relation to 2-UQ
2-UU	$u^2 = 1 + n,$ $n \in Nil(R)$	Always contained
2-UJ	$u^2 = 1 + j,$ $j \in J(R)$	Always contained
2-UQ	$u^2 = 1 + q,$ $q \in QN(R)$	Maximal in this chain

The flexibility of quasi-nilpotents makes 2-UQ rings encompass a broader class while still preserving substantive structural consequences when combined with potency.

3. Characterization and Structural Results for Potent 2-UQ Rings

The decisive result for potent 2-UQ rings is the following equivalence (Najafi et al., 14 Sep 2025):

$R$ is potent and 2-UQ,
$\iff$ $R/J(R)$ is tripotent,
$\iff$ $R$ is 2-UJ,
$\iff$ every element in $R$ can be written as $a = t + y$ where $t$ is tripotent ( $t^3=t$ ) and $y \in J(R)$ .

This characterization situates the potent 2-UQ property as a "semi-tripotency" condition—meaning the entire structure of the ring is controlled (modulo radical) by the tripotency identity $a^3 = a$ .

In these rings, the Jacobson radical $J(R)$ coincides with the set of quasi-nilpotent elements $QN(R)$ (Danchev et al., 23 Feb 2024), which further restricts pathological or degenerate behavior: for example, potent UQ rings are Dedekind finite, and $R/J(R)$ is Boolean under certain regularity assumptions.

4. Example Constructions and Behavior Under Extensions

A range of classical constructions are preserved under the potent 2-UQ property, provided certain conditions are met:

Trivial extensions, triangular matrix rings, power series extensions: The 2-UQ property is preserved if and only if the base ring possesses the property (Najafi et al., 14 Sep 2025).
Morita context rings: If trace ideals are nilpotent and central, the Morita context ring is 2-UQ if and only if each base ring is 2-UQ.
Group rings: Complete characterizations are established in (Najafi et al., 14 Sep 2025): If $2 \in J(R)$ and $RG$ is 2-UQ, then $G$ must be a $2$-group; if $3 \in J(R)$ and $RG$ is 2-UQ with $G$ a $p$ -group, then either $G$ is a $3$-group or $G$ has exponent $2$.

The result for group rings restricts potential examples sharply, showing that the potent 2-UQ property severely limits the allowable structure of $G$ .

Table: Behavior of 2-UQ under Constructions

Construction	Condition for Preservation
Trivial extension	Base ring is 2-UQ
Morita context	Corners 2-UQ; trace ideals nilpotent
Group ring $RG$	$G$ -group restricted by $J(R)$

These closure properties indicate that potent 2-UQ rings inherit their defining structural features under numerous standard ring-theoretic manipulations.

5. Connection with Regularity, Cleanness, and Tripotency

Potent 2-UQ rings display strong interactions with regularity and cleanness:

In regular rings, the 2-UQ property equivalently characterizes tripotency: regular, $\pi$ -regular reduced, unit regular, and tripotent are all equivalent in the presence of 2-UQ (Najafi et al., 14 Sep 2025, Mahmood et al., 8 Aug 2025).
For semi-potent rings, 2-UQ, 2-UJ, and 2-UU coincide modulo the radical and ensure tripotency in $R/J(R)$ .
Clean 2-UQ rings are necessarily semi-tripotent; if idempotents lift modulo $J(R)$ , the ring is clean.

This suggests a unification of several classical concepts—tripotency, cleanness, and various regularity conditions—all mediated by unit-square/quasi-nilpotence conditions when combined with potency.

Explicit examples demonstrate that 2-UQ, 2-UJ, and 2-UU are distinct classes:

In $A = \mathbb{F}_3\langle x, y: x^2 = 0 \rangle$ , $A$ is 2-UQ but not 2-UJ, as $J(A) = 0$ and yet some $u$ have $u^2 \in 1 + QN(A)$ (Najafi et al., 14 Sep 2025).
In $B = \mathbb{F}_2[[x]]$ , $B$ is 2-UQ but not 2-UU, since $(1+x)^2 = 1 + x^2$ is not nilpotent. A plausible implication is that while the 2-UQ property generalizes both 2-UJ and 2-UU, its boundary depends sensitively on the structure of $J(R)$ and $QN(R)$ . The absence of full symmetry indicates the nontriviality of the class and underscores the necessity for explicit structural characterization as provided above.

7. Connections to Quadratic Rings and Perfect Numbers

The paper of 2-powerfully perfect numbers in imaginary quadratic rings with unique factorization, notably $\mathcal{O}_{\mathbb{Q}(\sqrt{-1})}$ , $\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}$ , and $\mathcal{O}_{\mathbb{Q}(\sqrt{-7})}$ , creates direct analogies with potent 2-UQ phenomena (Defant, 2014). In these quadratic rings, the sum of 2-powered divisor norms equals twice the norm if and only if the element has a certain form reminiscent of classical perfect number theory but nuanced by ramification and splitting behavior of primes. The potent 2-UQ perspective in this context helps clarify the algebraic transfer of perfect number properties from $\mathbb{Z}$ to more general rings, and reveals how "potency" (as associated with lifting idempotents and structural regularity) underlies the arithmetic formulae for generalized perfect numbers.

Summary

Potent 2-UQ rings are semi-tripotent rings in which every unit $u$ satisfies $u^2 = 1 + q$ with $q$ quasi-nilpotent, and the radical and the set of quasi-nilpotents coincide. These rings generalize several unit- and radical-based ring classes, are preserved under many key constructions, force strict group-theoretic constraints in group rings, and unify classical tripotency, cleanness, and regularity. Recent work has produced a comprehensive framework, detailed closure properties, and explicit boundary examples, and extends the analogy to perfect numbers in certain quadratic rings, providing a fertile ground for further structural and homological investigations in modern ring theory (Najafi et al., 14 Sep 2025, Mahmood et al., 8 Aug 2025, Defant, 2014, Danchev et al., 23 Feb 2024, Hasanzadeh et al., 2 Jan 2025, Danchev et al., 2023).