Uniquely Strongly Clean Rings

• Uniquely strongly clean rings are defined by the unique decomposition of every element as the sum of a commuting idempotent and a unit, ensuring precise structural properties.
• The centrality of idempotents in these rings leads to Boolean quotients and plays a key role in classifying triangular and extension constructions.
• Applications include studies in matrix and group rings, although full matrix rings of order ≥2 often fail the uniqueness requirement due to decomposition constraints.

A uniquely strongly clean ring is a ring in which every element possesses a unique decomposition as the sum of a commuting idempotent and a unit. The uniquely strongly clean property introduces a high degree of algebraic rigidity, closely intersecting the structure theory of rings, the behavior of idempotents, unit groups, and clean-like decompositions. This property, and its generalizations, inform the classification of rings possessing canonical decompositions and their associated uniqueness phenomena.

1. Definitions and Conceptual Framework

Uniquely strongly clean rings are defined by the following condition: for every element $a \in R$, there exists a unique idempotent $e$ in the commutant of $a$ ($$e \in \operatorname{comm}(a) = \{x \in R \mid xa = ax\}$$), such that $a-e \in U(R)$, where $U(R)$ is the group of units of $R$, and $ae = ea$.

This property refines the classical concept of strongly clean rings (where each element is a sum $a=e+u$ with $e^2=e$ and $u \in U(R)$, $eu=ue$) by demanding uniqueness of $e$ for each $a$. In settings where all idempotents are central, the distinction between uniquely clean and uniquely strongly clean disappears.

Generalizations

• $n$-Strongly Clean Rings: Every element admits a decomposition as the sum of an idempotent and $n$ units commuting with the idempotent ($a = e + u_1 + \cdots + u_n$ with $e u_i = u_i e$). The $\Sigma$-strongly clean (E-strongly clean) rings generalize this further, requiring only existence of such a decomposition for some $n$.
• Strongly $J$-Clean Rings: Every element is the sum of an idempotent and a Jacobson radical element commuting with each other; the involutive variant (with projections) aligns these with uniquely strongly clean rings in the $*$-ring setting (Chen et al., 2012).
• Strongly $P$-Clean Rings: Decomposition into commuting idempotent and prime radical element; uniqueness arises precisely when idempotents are central (Chen et al., 2013).

2. Structural Conditions and Classification

Uniquely strongly clean rings exhibit several salient structural properties and often arise under strict constraints:

• Centrality of Idempotents: Uniqueness of decomposition enforces that all idempotents are central ((Chen et al., 2013), Theorem 1; (Danchev et al., 4 Jan 2024), Theorem 2.44; (Moussavi et al., 25 May 2025), Theorem 3.2). In abelian rings, uniquely special clean, uniquely strongly clean, and uniquely clean properties coincide.
• Boolean Quotients: For abelian (uniquely strongly clean) rings, the quotient $R/J(R)$ is Boolean, implying every element is idempotent ($x^2 = x$). This appears in the context of strong $P$-cleanness, strong $J$-cleanness, and perfect $J$-cleanness (Chen et al., 2013, Chen et al., 2013).
• Nilpotence or Potency: Potent rings (idempotents lift modulo $J(R)$, and $R/J(R)$ is Boolean or torsion) are central to the equivalence between uniquely strongly clean and other unique decomposability properties ((Danchev et al., 7 Jan 2024), Theorem 3.7; (Chen, 2014)).
• Jacobson Radical and Decomposable Structure: Additional requirements include $J(R)$ being nilpotent or locally nilpotent (cf. strong $P$-cleanness), and lifting properties of idempotents ((Chen et al., 2013); (Danchev et al., 4 Jan 2024), Theorem 2.12).

3. Behavior under Ring Constructions

Triangular and Matrix Rings

The uniquely strongly clean property is preserved in triangular matrix rings under suitable conditions. Specifically:

• Triangular Matrix Rings ($T_n(R)$): $T_n(R)$ is uniquely strongly clean if and only if $R$ is abelian (all idempotents central) and either $T_n(R)$ is uniquely strongly clean for some $n\geq1$ or every $n\geq1$ ((Chen et al., 2013), Theorem 1). In the commutative case, explicit unique decompositions can be constructed ((Chen et al., 2013), Theorem 9).
• Matrix Rings ($M_n(R)$): Full matrix rings with $n\geq2$ typically fail to be uniquely strongly clean or strongly $\Delta$-clean due to augmentation issues in the unit group ((Moussavi et al., 25 May 2025), Proposition 2.21; (Bergman, 20 Aug 2025), Example).
• Trivial Extensions and Morita Contexts: Preservation of uniqueness depends on underlying side rings being uniquely strongly clean and either local or abelian ((Moussavi et al., 25 May 2025), Theorems 4.2, 4.4; (Danchev et al., 4 Jan 2024)).

Ideals, Extensions, and Group Rings

• Homomorphic Images: The property of being uniquely strongly clean may not always pass to homomorphic images—this remains an open question ((Bergman, 20 Aug 2025), Question 19).
• Group Rings: For locally finite $p$-group $G$, the group ring $RG$ is uniquely strongly clean if and only if $R$ is uniquely strongly clean and $p \in J(R)$ ((Danchev et al., 6 Jan 2024), Theorem 3.3; (Moussavi et al., 25 May 2025), Theorem 5.2).

4. Relationship to Clean-Like and “Special” Ring Classes

Unique strong cleanness interacts closely with other clean-like properties:

• Special Clean, Special Almost Clean: In abelian rings, unit-regularity is equivalent to uniquely special clean; Rickart rings (annihilators generated by idempotents) equate to uniquely special almost clean ((Akalan et al., 2013), Proposition 4.1, Theorem 3.1).
• Perfectly Clean and Perfectly $J$-Clean Rings: These coincide with strong cleanness (and uniquely strongly clean) in matrix rings over local rings under weak cobleaching ((Chen et al., 2013), Theorem 3.2, Corollary 3.3, Proposition 4.4).
• Strongly $\Delta$-Clean Rings: Strong $\Delta$-cleanness refines the strongly clean class, particularly when idempotents are central, yielding uniqueness of decomposition ((Moussavi et al., 25 May 2025), Theorem 3.2 and 3.4).

5. Examples, Non-examples, and Limitations

The uniquely strongly clean property is highly restrictive:

• Examples:
  • Boolean rings.
  • Triangular matrix rings over Boolean rings or local rings with $R/J(R)\cong \mathbb{Z}_2$ ((Chen et al., 2013), Corollary 8).
  • Certain D-rings and group rings over uniquely strongly clean rings.
  • Strongly $J$-*-clean $*$-rings (equivalent to uniquely strongly $*$-clean) ((Chen et al., 2012), Theorem 3.2).
• Non-examples:
  • Full matrix rings $M_n(R)$ for $n\geq2$.
  • Non-abelian rings, or rings over fields with more than two elements ((Bergman, 20 Aug 2025), Example: $M_2(\mathbb{Z}/2\mathbb{Z})$ is strongly clean but not uniquely strongly clean).
  • Polynomial rings $R[x]$ and $R(x)$ over commutative rings fail to be $\Sigma$-strongly clean ((Singh, 2012), Example 1.16).

A plausible implication is that the existence of non-central idempotents, too many units, or the failure of $R/J(R)$ to be Boolean typically obstructs uniqueness.

6. Uniqueness, Centrality, and Connections to Involution

• Central idempotents: Uniqueness of decomposition enforces centrality of idempotents—if all idempotents are central, uniquely clean and uniquely strongly clean properties coincide ((Chen et al., 2013); (Bergman, 20 Aug 2025)).
• Involutive Rings: In the context of $*$-rings, the uniquely strongly clean condition is realized via strongly $J$-* clean rings, where the unique idempotent required in the decomposition is a projection ((Chen et al., 2012), Theorem 3.2).
• Annihilator Conditions: Uniqueness arguments often leverage annihilator properties (e.g., $aR \cap eR = \{0\}$) and exchange phenomena ((Akalan et al., 2013), Theorem 3.1; (Moussavi et al., 25 May 2025), Corollary 3.9).

7. Open Problems and Research Directions

Several unresolved or emerging areas include:

• Dedekind-finiteness: Whether uniquely strongly clean rings are Dedekind-finite remains unresolved—examples suggest strong cleanness does not guarantee Dedekind-finiteness (Bergman, 20 Aug 2025).
• Homomorphic Images: Does the property pass to homomorphic images? ((Bergman, 20 Aug 2025), Question 19).
• Full Characterization in Extension Settings: Classification for various ring extensions, ideals, and more general module categories is ongoing (Danchev et al., 4 Jan 2024, Moussavi et al., 25 May 2025).
• Matrix and Polynomial Rings: Determining necessary and sufficient conditions for matrix, triangular, or polynomial rings to be uniquely strongly clean (Chen et al., 2013, Moussavi et al., 25 May 2025).
• Interplay with Torsion and Power Conditions: Uniquely $\pi$-clean (uniquely $T$-clean) rings form a subclass characterized via torsion and central lifting (Chen, 2014).

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Uniquely strongly clean rings emerge as a confluence of strong decomposability, uniqueness, and centrality properties. Their paper informs foundational aspects of ring theory, including the finer structure of direct products, homomorphic images, matrix and triangular extensions, group rings, and involutive constructions. The uniqueness constraint yields a class that is highly regular in its idempotent/unit structure and tightly related to Boolean rings, local rings, and abelian group-theoretic conditions. These features situate uniquely strongly clean rings as a critical subclass of strongly clean rings, motivating ongoing investigations into their algebraic characterization and categorical extensions.