Strongly Clean Rings: Structure & Applications

• Strongly clean rings are unital rings where every element can be expressed as the sum of a commuting idempotent and a unit, establishing a robust framework for ring decomposition.
• Extensions such as strongly *-clean, strongly J-clean, and strongly P-clean rings introduce nuances including involution and nilpotent behavior that further refine these structural properties.
• Applications of strongly clean ring theory span matrix rings, triangular rings, group rings, and operator algebras, linking algebraic decompositions with module-theoretic and regularity properties.

A strongly clean ring is a unital ring in which every element is the sum of a commuting idempotent and unit; this property and its variants generate a rich structure theory at the interface of ring decompositions, regularity, and module-theoretic properties. The paper of strongly clean rings and their extensions—including *-clean rings (in the presence of involution), strongly J-clean and P-clean rings (involving radical or nilpotent summands), and a variety of generalizations—has uncovered deep algebraic connections, explicit structural characterizations, and important counterexamples. These insights extend to matrix and triangular matrix rings, group rings, and operator algebras, and are central to modern ring theory.

1. Formal Definitions and Foundational Results

Let $R$ be a unital associative ring.

• Strongly clean element: $a \in R$ is called strongly clean if there exists $e = e^2 \in R$ and $u \in U(R)$ (the unit group) such that $a = e + u$ and $eu = ue$.
• Strongly clean ring: $R$ is strongly clean if every $a \in R$ is strongly clean.

In the presence of an involution $*: R \to R$, one defines:

• Projection: $p \in R$ is a projection if $p^2 = p = p^*$.
• Strongly *-clean element: $a \in R$ is strongly *-clean if $a = p + u$ with $p$ a projection, $u \in U(R)$, and $pu = up$ (Chen et al., 2011).

Table 1 summarizes basic relationships between clean and strongly clean concepts:

Notion	Decomposition	Commutativity required	Involution constraint
clean	$a = e + u$	no	none
strongly clean	$a = e + u$	$eu = ue$	none
*-clean	$a = p + u$	no	$p^2 = p = p^*$
strongly *-clean	$a = p + u$	$pu = up$	$p^2 = p = p^*$

Every strongly clean ring is clean, and every strongly *-clean ring is strongly clean (but not conversely). In the *-setting, $R$ is strongly *-clean if and only if $R$ is strongly clean and every idempotent is a projection ($P(R) = \operatorname{Id}(R)$) (Chen et al., 2011).

2. Generalizations and Extensions

Several classes generalizing the strongly clean property have been introduced:

n-Strongly clean and Σ-strongly clean rings

• n-strongly clean: An element $x \in R$ is n-strongly clean if $x = e + u_1 + \dots + u_n$, with $e^2 = e$, each $u_i \in U(R)$, and $eu_i = u_ie$ for all $i$ (Singh, 2012).
• Σ-strongly clean: An element is Σ-strongly clean if it has an n-strongly clean decomposition for some $n$. The rings of integers $\mathbb{Z}$ are Σ-strongly clean but not n-strongly clean for any fixed $n$ (Singh, 2012).

Strongly nil-*, J-, and P-clean rings

• Strongly nil-*-clean: In a *-ring, every element is the sum of a commuting projection and a nilpotent (Chen et al., 2012). This is equivalent to every idempotent being a projection, $R$ periodic, and $R/J(R)$ Boolean.
• Strongly J-clean: Every element decomposes as $a = e + w$, with $e^2 = e$, $w \in J(R)$, and $e w = w e$. In formal matrix rings, precise root conditions on characteristic equations determine J-cleanness (Gurgun et al., 2013, Abdolyousefi et al., 2014).
• Strongly P-clean: Every element is the sum of an idempotent and a commuting strongly nilpotent element (element of the prime radical $P(R)$) (Chen et al., 2013). For local rings, strongly P-clean is equivalent to $R/J(R) \cong \mathbb{Z}_2$ and $J(R)$ locally nilpotent.

Strongly Δ-clean

• Strongly $\Delta$-clean (A-clean): Every element $a$ can be written $a = e + d$, $e^2 = e$, $ed = de$, with $d$ in $\Delta(R)$, the maximal unit-invariant subring of the Jacobson radical (Moussavi et al., 25 May 2025). Strongly $\Delta$-clean rings are always strongly clean; if idempotents are central, the uniquely clean condition is recovered.

Strongly NUS-nil clean

• Strongly NUS-nil clean: For every non-unit $a$, $a = e + n$ with $e^2 = e^4$, $n$ nilpotent, and $en = ne$. This is equivalent to $a^4 - a^2 \in \operatorname{Nil}(R)$ for all $a \notin U(R)$ (Doostalizadeh et al., 2 Aug 2025).

3. Structure Theorems, Matrix Rings, and Factorizations

A major research theme is the characterization of (strongly) clean elements in matrix, triangular, and group rings, and the lifting of cleanness properties from base rings to extensions.

Matrix and Triangular Matrix Rings

• Projective-free rings: For a projective-free commutative ring $R$ and a monic $h \in R[t]$, every $A \in M_n(R)$ with characteristic polynomial $h$ is strongly clean iff the companion matrix $C_h$ is strongly clean iff $h$ has a factorization $h = h_0 h_1$, $h_0$ invertible at $0$, $h_1$ invertible at $1$, and $\gcd(h_0, h_1) = 1$ (Chen et al., 2013).
• Commutative clean rings: Strongly clean matrix criteria extend to general commutative clean rings using Pierce sheaf techniques and gSRC-factorizations, with similarity of results to the local case (Burgess, 2014).
• Triangular matrix rings: For $T_n(R, \sigma)$, the skew triangular ring over a local $R$, $T_2(R, \sigma)$ is strongly clean iff certain $R$-module maps $l_a - r_{\sigma(b)}$ are surjective for $a$ near $1$ and $b$ in $J(R)$. For $T_3(R, \sigma)$, three related surjectivity maps must be surjective (Chen et al., 2013).

Characterizations by Root Criteria and Factorizations

• Strongly J-clean 2x2 matrices: Over 2-projective-free rings, a matrix is strongly J-clean iff it is similar to a matrix $$[\begin{smallmatrix}0 & \lambda \ 1 & \mu\end{smallmatrix}]$$ with $\lambda \in J(R)$, $\mu \in 1+J(R)$, and the quadratic $x^2 - \mu x - \lambda = 0$ has one root in $J(R)$, one in $1+J(R)$ (Abdolyousefi et al., 2014).
• Strongly J$\sharp$-clean matrices: For projective-free rings, a matrix is strongly J$\sharp$-clean if the characteristic polynomial admits SR-type factorizations modulo $J\sharp(R)$ (Chen et al., 2014).

Abstract Structure and Relationships

• Strongly clean (and closely related uniquely clean, strongly J-clean, etc.) rings are typically quasi-duo with $R/J(R)$ Boolean if certain radical conditions hold (Moussavi et al., 25 May 2025, Chen et al., 2013).
• Central idempotents and idempotent lifting modulo $J(R)$ play crucial roles in distinguishing uniquely clean and strongly clean classes (Danchev et al., 7 Jan 2024).

4. Interrelations and Hierarchies

Explicit implications and proper inclusions between the main classes are illustrated in the diagram below:

Ring property	Implies	Reverse holds?
strongly *-clean	strongly clean	Only if $P(R) = \operatorname{Id}(R)$ (Chen et al., 2011)
strongly nil-* clean	strongly *-clean	No
strongly J-clean	strongly clean	No
strongly P-clean	strongly nil-clean	No
strongly Δ-clean	strongly clean	Yes
strongly NUS-nil clean	strongly clean	No

For GUSC (generalized uniquely strongly clean) and closely associated classes:

Class	Definition
USC	All elements uniquely strongly clean
GUSC	Non-units uniquely strongly clean
GUC	Non-units uniquely clean
CUSC	Clean elements uniquely strongly clean
CUC	Clean elements uniquely clean

USC $\implies$ GUSC $\implies$ strongly clean (Danchev et al., 6 Jan 2024); every CUSC and potent ring is USC (Danchev et al., 7 Jan 2024).

5. Special and Limiting Constructions

Operator Algebras and von Neumann Algebras

It is proven that all finite von Neumann algebras and separable infinite factors are clean rings, with uniform norm control on the inverses involved in the decomposition (Cui et al., 2021). The strongly clean property in this context is subtler and remains a subject of further investigation.

Embedding and Extensions

Open questions examine whether every ring embeds in a strongly clean ring, or whether every strongly clean ring is Dedekind-finite. A new example constructs a ring $R$ containing strongly clean elements $x, y$ with $xy = 1$, $yx \neq 1$, suggesting a possible negative answer to the Dedekind-finiteness problem for strongly clean rings (Bergman, 20 Aug 2025).

In the field of homomorphic images and extensions, uniquely strongly clean properties are generally preserved under homomorphic images when idempotents are central (Bergman, 20 Aug 2025).

6. Key Examples, Counterexamples, and Classification Criteria

Concrete constructions serve both as positive classification tools and as explicit counterexamples:

• Clean but not *-clean: $R = \mathbb{Z}_2 \oplus \mathbb{Z}_2$ with involution $(a,b)^* = (b,a)$ is strongly clean but not strongly *-clean because $P(R) \ne \operatorname{Id}(R)$ (Chen et al., 2011).
• 2-strongly clean but not strongly clean: $R = \mathbb{Z}_p G$ with $p\ne 2$, $G$ cyclic of order 3, is 2-strongly clean but not strongly clean (Singh, 2012).
• Strongly clean but not strongly J-clean: Examples in formal matrix rings and certain group rings demarcate these classes (Gurgun et al., 2013, Abdolyousefi et al., 2014).
• Triangular matrix extension: For a commutative ring $R$, $T_n(R)$ is perfectly J-clean iff $R$ is strongly J-clean; in many cases, perfect and strongly clean coincide in matrix and triangular matrix contexts (Chen et al., 2013).
• Central Idempotents and Matrix Rings: Strongly clean property can break down in matrix rings unless additional root or factorization conditions are met (Chen et al., 2013, Burgess, 2014).

7. Implications, Open Problems, and Future Directions

The paper of strongly clean and related decomposition properties continues to motivate structural exploration, especially in connection to:

• Refined radical and regularity conditions (e.g., $\Delta$-clean, P-clean, J-clean).
• Matrix and module-theoretic transfer of strong cleanness.
• Extensions and factor rings, particularly the behavior under Morita equivalence and formation of trivial/group/triangular extensions.
• The interplay with the Dedekind-finiteness property, which is unresolved in the context of strongly clean rings and is the subject of new counterexamples (Bergman, 20 Aug 2025).
• Characterization and uniqueness properties (GUSC, CUSC, potent rings), especially under centrality constraints on idempotents (Danchev et al., 7 Jan 2024, Danchev et al., 6 Jan 2024, Moussavi et al., 25 May 2025).

A plausible direction for further research is the universal characterization of strongly clean properties in operator algebraic settings, the precise distinction and overlap with nil-clean and P-clean phenomena in noncommutative matrix rings, and the paper of transfer properties in group and skew group rings under various nilpotence and centrality assumptions.

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This article synthesizes recent advances and central results, referencing (Chen et al., 2011, Singh, 2012, Chen et al., 2012, Chen et al., 2013, Chen et al., 2013, Chen et al., 2013, Chen et al., 2013, Chen et al., 2013, Gurgun et al., 2013, Burgess, 2014, Chen et al., 2014, Abdolyousefi et al., 2014, Cui et al., 2015, Cui et al., 2021, Danchev et al., 6 Jan 2024, Danchev et al., 7 Jan 2024, Moussavi et al., 25 May 2025, Doostalizadeh et al., 2 Aug 2025), and (Bergman, 20 Aug 2025). The field remains active, with substantial interplay among decomposition theory, module structure, and abstract algebraic properties.