Strongly NUS-nil Clean Rings

• Strongly NUS-nil clean rings are rings where every non-unit element can be expressed as the sum of a nilpotent element and a square-idempotent that commute, characterized by a^4-a^2 being nilpotent.
• They occupy an intermediate position between strongly nil-clean and strongly clean rings, exhibiting distinct decomposition properties in matrix, triangular, and group ring constructions.
• Structural criteria such as a nil Jacobson radical and local ring conditions underscore their importance in understanding module theory and ring decompositions.

A strongly NUS-nil clean ring is a ring in which every non-unit element can be expressed as the sum of a nilpotent element and a square-idempotent (an element e such that $e^2 = e^4$), with the additional requirement that these two components commute. This class of rings provides an intermediate structure between strongly nil-clean rings (where every element is the sum of a commuting idempotent and nilpotent) and strongly clean rings (where every element is a commuting sum of a unit and an idempotent). The algebraic and structural richness of strongly NUS-nil clean rings is illuminated through precise polynomial identities, decomposability criteria, results about Jacobson radicals, and their behavior with respect to various ring-theoretic constructions, including matrix and group ring extensions (Doostalizadeh et al., 2 Aug 2025).

1. Defining Properties and Core Characterization

The primary defining property is that for any non-unit $a \in R$,

• there exist elements $e \in R$ (square-idempotent: $e^2 = e^4$) and $n \in Nil(R)$ (nilpotent: $n^k = 0$ for some $k$), such that:
  • $a = e + n$,
  • $en = ne$.

A central algebraic characterization is that a ring $R$ is strongly NUS-nil clean if and only if for every $a \notin U(R)$,

$a^4 - a^2 \in Nil(R).$

This provides an effective polynomial identity criterion for verifying the property in specific rings. Occasionally, the slightly weaker condition $a^3 - a \in Nil(R)$ may also be deduced for nonunits.

The use of square-idempotent elements encompasses all classical idempotents ($e^2 = e$) but allows for the inclusion of more general elements, which broadens the class beyond strongly nil-clean rings.

2. Position Within the Landscape of "Clean" Rings

Strongly NUS-nil clean rings are strictly situated between strongly nil-clean rings and strongly clean rings:

Property	Every Element	Non-Units Only	Idempotent Type	Clean Part Type
Strongly nil-clean	Yes	-	Idempotent ($e^2=e$)	Nilpotent
Strongly NUS-nil clean	No	Yes	Square-idempotent	Nilpotent
Strongly clean	Yes	-	Idempotent ($e^2=e$)	Unit

Every strongly nil-clean ring is strongly NUS-nil clean, and every strongly NUS-nil clean ring is strongly clean ($a^4-a^2 \in Nil(R)$ can force $a$ to be almost idempotent modulo nilpotents, so their clean decompositions are tightly controlled). However, there exist strongly clean rings which are not strongly NUS-nil clean (Doostalizadeh et al., 2 Aug 2025); similarly, the decomposition for strongly NUS-nil clean rings is not required for units.

3. Structural Criteria and Consequences

The fundamental structure theorem states that $R$ is strongly NUS-nil clean if and only if for every non-unit $a \in R$, $a^4 - a^2 \in Nil(R)$.

Further, if a ring $R$ has only trivial idempotents (i.e., $0$ and $1$), then $R$ is strongly NUS-nil clean if and only if $R$ is a local ring with nil Jacobson radical $J(R)$ (i.e., $J(R)$ consists of nilpotent elements). This is a particularly sharp characterization: in such rings, all non-units are automatically contained in the Jacobson radical, which is nil.

For any strongly NUS-nil clean ring, the Jacobson radical $J(R)$ is always nil. The property is also local with respect to nil ideals: if $I \subset R$ is a nil ideal, then $R$ is strongly NUS-nil clean if and only if $R/I$ is so.

This criterion is robust under certain ring constructions; for example, for corners $eRe$ (where $e$ is idempotent) and suitable extensions or reductions, the strongly NUS-nil clean property can be inherited.

4. Behavior in Matrix Rings, Triangular, and Extension Constructions

Strongly NUS-nil clean rings have nuanced behavior with respect to matrix and extension constructions:

• Matrix rings: Examples show that $\mathbb{Z}_3 \times \mathbb{Z}_3$, $M_2(\mathbb{Z}_3)$, and $M_2(\mathbb{Z}_2)$ are strongly NUS-nil clean without being strongly nil-clean; however, $M_n(R)$ for $n \geq 3$ generally fails to be strongly NUS-nil clean. This reflects the fact that nil-cleanness or its variants do not always extend to full matrix rings of large size.
• Triangular and skew triangular matrix rings: $T_n(R)$ is strongly NUS-nil clean for all $n$ if and only if $R$ is strongly square-nil clean. The same equivalence holds for certain skew triangular matrix rings $T_n(R,\alpha)$.
• Trivial extensions and Morita contexts: The trivial extension $T(R,M)$ of $R$ by an $(R,R)$-bimodule $M$ is strongly NUS-nil clean if and only if $R$ is. More generally, in Morita context rings with nil trace ideals, the property is governed by the property on both corner rings $A$ and $B$.
• Group rings: For a locally finite $p$-group $G$ and a ring $R$ with $p \in J(R)$ nilpotent, the group ring $RG$ is strongly NUS-nil clean if and only if $R$ is strongly NUS-nil clean and the augmentation ideal is nil.

These results provide a range of examples and counterexamples and establish the limits within which the property can be expected to be preserved under classical ring-theoretical constructions.

5. Implications for Ring Structure and Module Theory

The central polynomial criterion $a^4 - a^2 \in Nil(R)$ for non-units gives structural insight into the "nilpotency proximity" of non-units to square-idempotent elements. In rings with only trivial idempotents, this proximity is so strong that all non-units must be nilpotent, hence the equivalence with local rings having nil Jacobson radicals.

The positioning of strongly NUS-nil clean rings between strongly nil-clean and strongly clean rings enables a refined understanding of the clean behavior of elements, offering a finer subdivision of the landscape of "clean-type" rings. This can yield further information about exchange properties, the lifting of idempotents, decompositions of modules, and module-theoretic behavior related to the decomposition of elements in associated module categories.

These decompositions have implications for the paper of modules over such rings—specifically, continuous, exchange, or projective modules—because the ring's internal cleanness often reflects or constrains module behavior, such as direct sum decompositions or lifting properties.

6. Prospects for Further Research and Broader Context

The paper of strongly NUS-nil clean rings introduces a robust algebraic criterion in the form of $a^4 - a^2 \in Nil(R)$, extending the taxonomy of rings governed by decompositional properties. Their precise placement within the hierarchy of clean-type rings informs both ring-theoretic classification problems and construction of new examples through matrix rings, group rings, and extensions.

Potential future research includes:

• Further identification of ring classes and explicit examples (and counterexamples) for the property in less classical settings, such as nonassociative or infinite-dimensional rings.
• Analysis of the behavior under less restrictive types of extensions, e.g., non-nil or infinite-dimensional module extensions.
• Examination of the interaction with additional ring-theoretic properties such as exchange, chain conditions, and representation type.
• Applications in module theory and representation theory, leveraging the controlled decompositions available in large rings constructed from strongly NUS-nil clean rings.

The class of strongly NUS-nil clean rings thus serves both as an object of intrinsic interest and as a tool for constructing and understanding more intricate algebraic structures (Doostalizadeh et al., 2 Aug 2025).