A Generalization of UQ Rings

Abstract: We examine the newly defined class of {\it $n$-$UQ$ rings} described by the condition that $uⁿ - 1 \in QN(R)$ for every unit $u \in U(R)$, where $QN(R)$ denotes the set of quasi-nilpotent elements (see \cite{Tien}). This class naturally extends the recently defined class of rings in \cite{daoa} and \cite{dam}, as well as expectedly generalizes previously explored concepts such as $UJ$, $UU$ and $UQ$ rings. We conduct here a comprehensive structural analysis of these $n$-$UQ$ rings and study their stability under various ring-theoretic constructions including matrix rings, group rings, trivial extensions and power series rings. As a result, several new characterizations are established, thus revealing relevant connections between $n$-$UQ$ rings and fundamental classes of rings such as reduced, clean, exchange, semi-regular and potent rings, respectively. Moreover, we prove that the classes of $n$-$UJ$ and $n$-$UU$ rings are properly contained in the class of $n$-$UQ$ rings. These achievements not only unify and expand existing theories in this branch, but also provide a robust framework for possible further investigations into the interplay between the unit behavior and quasi-nilpotency in noncommutative ring theory.

• The paper establishes that n-UQ rings are closed under direct products, corners, suitable quotients, triangular constructions, trivial extensions, truncated polynomial rings, and formal power series.
• The paper proves that odd-index (2n−1)-UQ rings are Dedekind-finite and, in semi-potent or potent settings, links n-UQ, n-UJ, and n-UU properties to the identity x^{2n}=x modulo the Jacobson radical.
• The paper demonstrates that n-UQ strictly generalizes n-UU and n-UJ through explicit counterexamples, while group-ring transfer results and polynomial-ring criteria remain conditional and motivate open problems.

A Generalization of UQ Rings: Structure and Stability of $n$-UQ Rings

The class of $n$-UQ rings

The paper under review, by Danchev, Doostalizadeh, and Hasanzadeh (2606.15204), undertakes a systematic structural study of $n$-$UQ$ rings. A unital (not necessarily commutative) ring $R$ is defined to be $n$-$UQ$, for a fixed integer $n \geq 2$, if $u^n - 1 \in QN(R)$ for every unit $u \in U(R)$, where $n$0 denotes the set of quasi-nilpotent elements — those $n$1 for which $n$2 is invertible whenever $n$3 commutes with $n$4. Since the inclusions $n$5 always hold, every $n$6-$n$7 ring ($n$8) and every $n$9-$n$0 ring ($n$1) is automatically $n$2-$n$3. The concept refines the $n$4 rings introduced in prior work of Danchev et al., where $n$5, and extends the hierarchy UJ ⊂ UU ⊂ UQ to its "$n$6-th power" analogues.

The authors document that both reverse inclusions fail. Two explicit counterexamples anchor the theory: an $n$7-algebra generated by $n$8 subject only to $n$9, which is $UQ$0-$UQ$1 (hence $UQ$2-$UQ$3) but not $UQ$4-$UQ$5 since it is semiprimitive; and the formal power series ring $UQ$6, which is $UQ$7-$UQ$8 for all $UQ$9 but not $R$0, because $R$1 is a unit that is not unipotent. Their direct product is simultaneously $R$2-$R$3 while failing both stronger properties, so the containment chain is strict at each level.

Closure properties and matrix obstructions

The basic closure theory mirrors that of related classes. The property passes to arbitrary direct products componentwise, to corners $R$4, to rationally closed subrings, and to the center; moreover, if $R$5 and $R$6 is $R$7-$R$8, then so is $R$9 — a lifting result proved via stability of quasi-nilpotents modulo radical ideals.

Two negative results carry substantial structural weight. First, $n$0 is never $n$1-$n$2 for any $n$3: the rotation matrix $n$4 yields either a unit or the identity lying in $n$5, both contradictions. Second, as a consequence, every $n$6-$n$7 ring is Dedekind-finite, since a non-Dedekind-finite ring contains a corner isomorphic to some $n$8. This is a strong rigidity statement: odd-index $n$9-$UQ$0 rings cannot admit one-sided inverses that are not two-sided.

Stability under standard constructions is established cleanly:

Construction	Criterion
Trivial extension $UQ$1	$UQ$2 is $UQ$3-$UQ$4 iff $UQ$5 is
Formal triangular matrices	$UQ$6-$UQ$7 iff both diagonal rings are
Triangular matrix ring $UQ$8	$UQ$9-$n \geq 2$0 iff $n \geq 2$1 is
Quotient $n \geq 2$2	$n \geq 2$3-$n \geq 2$4 iff $n \geq 2$5 is
Power series $n \geq 2$6	$n \geq 2$7-$n \geq 2$8 iff $n \geq 2$9 is

For polynomial rings the situation is more delicate and requires the 2-primal hypothesis ($u^n - 1 \in QN(R)$0). Under that assumption, the four conditions — $u^n - 1 \in QN(R)$1 being $u^n - 1 \in QN(R)$2-$u^n - 1 \in QN(R)$3, and $u^n - 1 \in QN(R)$4 being respectively $u^n - 1 \in QN(R)$5-$u^n - 1 \in QN(R)$6, $u^n - 1 \in QN(R)$7-$u^n - 1 \in QN(R)$8, or $u^n - 1 \in QN(R)$9-$u \in U(R)$0 — are equivalent. This rests on the identity $u \in U(R)$1 valid for 2-primal rings, together with Chen's description of units of $u \in U(R)$2. Notably, the equivalence forces $u \in U(R)$3 to be $u \in U(R)$4-$u \in U(R)$5 only through nilpotency rather than genuine quasi-nilpotency; whether $u \in U(R)$6 can be $u \in U(R)$7-$u \in U(R)$8 over a non-2-primal base remains outside the scope of the paper.

Characterization in the semi-potent and potent settings

The central results concern rings with abundant idempotents. For a semi-potent ring $u \in U(R)$9 (every one-sided ideal outside $n$00 contains a non-zero idempotent), the following are equivalent: $n$01 is $n$02-$n$03; $n$04 satisfies the polynomial identity $n$05; $n$06 is $n$07-$n$08; and $n$09 is $n$10-$n$11. The proof proceeds by showing first that such a quotient must be reduced — otherwise Levitzki's theorem produces a $n$12 matrix corner, contradicting Proposition on matrix rings — and then that any failure of $n$13 yields a unit equation $n$14 inside a corner, which is impossible in an $n$15-$n$16 ring.

For potent rings (semi-potent plus idempotent lifting mod $n$17), the equivalence list expands to six conditions, adding that $n$18 itself is $n$19-$n$20 and $n$21 is $n$22-$n$23. Thus, in the potent setting, the $n$24-$n$25, $n$26-$n$27, and $n$28-$n$29 notions collapse into the single algebraic condition $n$30 modulo the Jacobson radical. This is the paper's strongest unifying claim, and it immediately implies several corollaries:

• For a $n$31-$n$32 ring, regularity, $n$33-regularity plus reducedness, strong regularity, unit-regularity, and the identity $n$34 are all equivalent.
• Exchange, clean, and semi-regular are equivalent within the class of $n$35-$n$36 rings — extending the known collapse for $n$37-$n$38 rings.
• Introducing the terminology semi-$n$39-potent (potent with $n$40 mod $n$41), a ring is a clean $n$42-$n$43 ring exactly when it is semi-$n$44-potent.
• For artinian (in particular finite) rings, $n$45-$n$46, $n$47-$n$48, and $n$49-$n$50 coincide — a context in which the strict hierarchies of the general theory disappear entirely.

Group rings

The group-ring section gives partial transfer results. If $n$51 is $n$52-$n$53, then $n$54 is $n$55-$n$56 (as a rationally closed subring); conversely, if $n$57 is $n$58-$n$59 and $n$60 is a locally finite $n$61-group with $n$62, then $n$63 is $n$64-$n$65, using the containment of the augmentation ideal $n$66 in $n$67 and the lifting corollary. A final lemma imposes a sharp restriction: if $n$68 is $n$69-$n$70 with $n$71 and $n$72 is a $n$73-group, then $n$74 has exponent $n$75. The argument exploits the unit $n$76 and the factorization $n$77 to force $n$78.

Limitations and open problems

The paper's characterizations are largely confined to odd exponents $n$79; the even case is treated only sporadically (e.g., the trivial-extension criterion holds for all $n$80, and the semi-potent theorem concerns odd indices specifically). The polynomial-ring equivalence requires the 2-primal assumption, and the group-ring results give only one-directional transfers except under restrictive hypotheses on the coefficient ring's radical. The authors explicitly pose two open problems: first, to determine necessary and sufficient conditions on $n$81 and $n$82 for $n$83 to be $n$84-$n$85; second, to find conditions under which the $n$86-$n$87 property (for $n$88) actually forces the $n$89 property — i.e., whether the hierarchy collapses back at the level of individual rings under suitable hypotheses.

Conclusion

This work consolidates the theory of $n$90-$n$91 rings by situating them precisely between the $n$92-$n$93/ $n$94-$n$95 classes and the broader quasi-nilpotent framework. Its principal contributions are the Dedekind-finiteness of odd-index $n$96-$n$97 rings, the complete stability analysis under matrix-type and analytic constructions, and the potent/semi-potent characterization reducing the entire hierarchy to the identity $n$98 modulo the Jacobson radical. The results demonstrate that, once enough idempotents are available, quasi-nilpotency contributes nothing beyond the Jacobson radical, while in the general case the class is genuinely wider than both $n$99-$n$00 and $n$01-$n$02 rings.