2-UQ Group Rings: Theory and Extensions

• 2-UQ group rings are defined as rings in which every unit squared equals an idempotent plus a quasi-nilpotent, generalizing the 2-UJ and 2-UU classes.
• They impose strict constraints on the underlying group, requiring groups to be 2-groups or 3-groups (or of exponent 2) under specific ring conditions.
• Their stability under extensions like Morita contexts, triangular, and trivial extensions connects them to tripotent, potent, and clean ring structures.

A 2-UQ group ring refers to a group ring $RG$ (with $R$ a ring, typically with unity) in which every unit %%%%2%%%% satisfies $u^2 = e + q$, where $e$ is an idempotent in $RG$ (often $e = 1$ for most natural examples) and $q \in QN(RG)$, the set of quasi-nilpotent elements, and $e$ and $q$ commute. The class is strictly more general than the previously studied 2-UJ and 2-UU rings, absorbing those as special cases. Research on 2-UQ group rings uncovers strong structural constraints on the underlying group $G$ and links to several other salient classes, including tripotent, potent, and clean rings.

1. Definition and Fundamental Properties

A ring $R$ is called a 2-UQ ring if for every unit $u \in U(R)$, one can write

$u^2 = e + q,$

where $e$ is an idempotent, $q \in QN(R)$ (quasi-nilpotent), and $e$ commutes with $q$. For group rings $RG$, quasi-nilpotency is understood as: for any $x \in RG$ commuting with $u$, $1 - qx$ is invertible.

This definition encompasses 2-UJ rings ($u^2 = 1 + j$, $j \in J(R)$) and 2-UU rings ($u^2 = 1 + n$, $n$ nilpotent), but not conversely. Every 2-UJ or 2-UU ring is 2-UQ, but counter-examples exhibit rings like $R = \mathbb{F}_3\langle x,y : x^2 = 0\rangle$, which are 2-UQ but not 2-UJ (since $J(R)=0$). The quasi-nilpotent set $QN(R)$ contains both $Nil(R)$ and $J(R)$.

2. Group Ring Structure and Constraints on $G$

The imposition of the 2-UQ property on $RG$ exerts tight restrictions on the group $G$. The principal characterizations established include:

• If $RG$ is a 2-UQ ring and $2 \in J(R)$, then $G$ must be a 2-group.
• If $3 \in J(R)$, $G$ is either a 3-group or a group of exponent 2 (i.e., $g^2 = 1$ for all $g$).

To see why, consider units $u$ built from $g \in G$ and analyze $(1-g^2) \in QN(RG)$. If $g$ has order not a power of 2 (respectively not 3), this produces a contradiction with the unit property and quasi-nilpotency.

Table: Constraints on $G$ in 2-UQ Group Rings

Condition on $R$	Consequence for $G$
$2 \in J(R)$	$G$ is a 2-group
$3 \in J(R)$	$G$ is 3-group or exp 2

These facts create a pronounced boundary for possible group rings attaining the 2-UQ property.

3. Relationship to 2-UJ, 2-UU, and Tripotent Rings

Every 2-UJ ring is 2-UQ because $J(R) \subseteq QN(R)$; similarly, every 2-UU ring is 2-UQ. However, not every 2-UQ ring is 2-UJ or 2-UU—explicit counter-examples exhibit that the inverse statements fail.

A notable connection is with tripotent rings ($R$ such that every element $a$ satisfies $a^3 = a$). The quotient $R/J(R)$ is tripotent precisely when $R$ is potent and 2-UQ. For potent rings, the following equivalences hold:

• $R$ is 2-UQ $\iff$ $R/J(R)$ is tripotent $\iff$ $R$ is 2-UJ.

For regular (strongly regular/unit regular) rings, being 2-UQ is equivalent to being tripotent, and a reduced regular 2-UQ ring is a subdirect product of fields $\mathbb{F}_2$ and $\mathbb{F}_3$.

4. Extensions: Morita Contexts and Other Constructions

The 2-UQ property is structurally stable under certain extensions:

• Morita Contexts: For a Morita context ring $$\left[ \begin{smallmatrix} A & M \ N & B \end{smallmatrix}\right]$$ with nilpotent central trace ideals $MN$ and $NM$, the ring is 2-UQ iff $A$ and $B$ are 2-UQ.
• Triangular and Matrix Rings: Triangular matrix rings and skew/ordinary power series over a 2-UQ ring are themselves 2-UQ given suitable constraints.
• Trivial Extensions: Trivial extensions of a 2-UQ ring by a bimodule retain the 2-UQ property.

These results establish Morita invariance (under nilpotent trace ideals) and extension-theoretic preservation for the 2-UQ property.

5. Interaction with Regularity, Potency, and Cleanness

In-depth analysis reveals that the 2-UQ property interacts closely with other ring-theoretic conditions:

• Regularity: A 2-UQ ring is regular iff it is tripotent.
• Cleanness: If $R$ is 2-UQ and clean, then $R$ is semi-tripotent; conversely, semi-tripotent rings are clean 2-UQ.
• Potency/Semi-potency: For potent rings, 2-UQ, 2-UJ, and $R/J(R)$ tripotent are equivalent. For semi-potent rings, $R/J(R)$ tripotent $\iff$ $R$ is 2-UJ.

Power series rings and triangular matrix extensions preserve the 2-UQ property under these conditions whenever the base ring does.

6. Comparison and Expansion of Previous Results

Earlier studies, notably by Cui-Yin (Commun. Algebra, 2020) and Danchev et al. (J. Algebra Appl., 2025), focused primarily on 2-UJ and strongly quasi-nil clean rings. The 2-UQ class formulated herein explicitly addresses structural boundary cases where rings are 2-UQ but not 2-UJ or 2-UU. Furthermore, the expansion to group rings and Morita contexts, and the demonstration of interactions with regularity, tripotency, and clean decompositions, yield a broader unifying perspective. The distinct constraint on underlying groups for 2-UQ group rings (2-groups or 3-groups/exp 2) is especially novel.

7. Key Examples and Counter-examples

• $A = \mathbb{F}_3\langle x, y \rangle$ with $x^2 = 0$ is 2-UQ but not 2-UJ, since every unit squares to $1$ plus a nilpotent, but $J(A)=0$.
• $B = \mathbb{F}_2[[x]]$ is 2-UQ but not 2-UU (since $(1+x)^2 = 1 + x^2 \notin 1 + Nil(B)$).
• Morita context rings with nilpotent trace ideals are 2-UQ precisely when their diagonal blocks are.

These examples illustrate the strict inclusions and provide boundaries between subclasses.

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In summary, 2-UQ group rings are characterized by the property that every unit squares to an idempotent plus a quasi-nilpotent, a condition generalizing prior notions and tightly controlling the allowable group structures for $RG$. This framework robustly connects with concepts of tripotency, regularity, clean and exchange structures, and exhibits stability under natural ring and module constructions. The explicit constraints on groups, ramifications for extensions, and firm links to well-known ring classes position 2-UQ group rings as central objects in modern ring theory (Najafi et al., 14 Sep 2025).