Star Operation Method in Noncommutative Ideal Theory

• Star Operation Method is a closure-type operator on ideals that defines divisorial closures and examines invertibility in noncommutative rings.
• It underpins the definition of right Prüfer ν-multiplication orders, extending classical multiplicative ideal theory to noncommutative domains.
• The method ensures overring stability, unifying and generalizing ideal-theoretic structures in both commutative and noncommutative settings.

A star operation is a closure-type operator on the ideals of a ring, particularly significant in the multiplicative ideal theory of both commutative and non-commutative rings. The "star operation method," as developed in the context of orders in simple Artinian rings, provides a versatile and robust tool for analyzing divisorial closures and generalized invertibility, leading to the non-commutative analogs of classical commutative structures such as Prüfer ν-multiplication domains. The approach centers on the use of special star operations—most notably the v-operation—to define and characterize right Prüfer ν-multiplication orders, their closure properties under overrings, and their utility in organizing the ideal-theoretic structure of non-commutative rings.

1. Formal Definition and Role of Star Operations

A star operation on an order $S$ in a simple Artinian ring $Q$ is a function $I \mapsto I^*$ from the set of nonzero right $S$-ideals (embedded in $Q$) that satisfies:

1. Unit invariance: $(uI)^* = uI^*$ for every unit $u \in Q$.
2. Order preservation: If $I \subseteq J$ then $I^* \subseteq J^*$.
3. Idempotence and extensivity: $I \subseteq I^*$ and $(I^*)^* = I^*$.

A principal example is the $v$-operation (termed the $\nu$-operation in some literature). For a right ideal $I$,

$I_v = (S : (S : I)_l)$

where $(S : I)_l = \{q \in Q : Iq \subseteq S\}$ is the left idealizer, capturing the divisorial closure within $Q$. This operation encodes crucial divisorial information, distinguishing itself in non-commutative settings via its asymmetric treatment of right versus left ideals. The focus on this operation is justified as it underlies the extension of Bezout-type, GCD-type, and Prüfer-type structures to the non-commutative regime.

2. Right Prüfer ν-Multiplication Orders

The main innovation is the definition of right Prüfer ν-multiplication orders, which generalize the classical notion of Prüfer $v$-multiplication domains (PvMDs) from commutative algebra.

Let $H_t(S)$ denote the set of finitely generated right ideals of finite type in $S$. Then $S$ is a right Prüfer ν-multiplication order if, for every $I \in H_t(S)$,

$$\bigl((S : I)_v I\bigr)' = S \ \bigl(I (S : I)_v\bigr)' = O_1(I)$$

where:

• $(S : I) = \{q \in Q : qI \subseteq S\}$ is the right idealizer.
• $(\,\cdot\,)'$ denotes star closure under the v-operation.
• $O_1(I)$ is the largest overring of $S$ in $Q$ where $I$ remains a right ideal.

These conditions represent a non-commutative avatar of invertibility: the closure equations serve as the generalization of $II^{-1} = R$ in the commutative case, now mediated by the v-operation and adapted to module-theoretic asymmetry. Right Prüfer ν-multiplication orders form a class strictly containing right v-Bezout orders (where all finitely generated ideals are principal in the v-sense) and right GCD-orders, thus enabling a unified treatment of divisibility and invertibility in a broader context.

3. Closure Properties and Overring Stability

A key structural result (Proposition 4.11) asserts that the property of being a right Prüfer ν-multiplication order is preserved under passage to overrings. If $T$ is an overring of $S$ (i.e., $S \subseteq T \subseteq Q$ and $T$ is also an order), and $S$ is right Prüfer ν-multiplication, then so is $T$. Explicitly, for any $I' \in H_t(T)$ obtained by extending $I \in H_t(S)$,

$$\bigl((T : I')I'\bigr)' = T \qquad \text{and} \qquad \bigl(I'(T : I')\bigr)' = O_1(I').$$

This stability under overrings generalizes classical facts about Prüfer domains and PvMDs and is crucial for the theory's applicability. It ensures that ideal-theoretical and multiplicative properties are robust under localizations and extensions, enabling classification via overring structure and facilitating local-global analyses.

4. Mathematical Characterizations and Proof Strategies

The effectiveness of the star operation method is underscored by concrete algebraic formulas and mechanisms. Central identities include:

• Divisorial closure: $I_v = (S : (S : I)_l)$.
• Prüfer ν-multiplication structure:

  $$\bigl((S : I)_v I\bigr)' = S, \qquad \bigl(I (S : I)_v\bigr)' = O_1(I).$$

Proofs utilize adaptation of multiplicative ideal theory from commutative to non-commutative contexts, leveraging divisorial structures and analyzing how star operations behave under localization and extension. For the overring closure result, the argument proceeds by taking a finitely generated ideal and verifying the translation of closure conditions through generator extension and recalculation of v-closures within the larger overring, with careful attention to the induced module and divisorial interrelations.

5. Classification and Applications

The theory has significant implications for the classification of orders in simple Artinian rings:

• Invariant structure: The star operation (especially the v-operation) serves as a multiplicative invariant, closely tied to divisoriality and invertibility properties of ideals.
• Unification: Prüfer ν-multiplication orders provide a framework encompassing v-Bezout orders, right GCD-orders, and can be used to analyze non-commutative analogs of Krull domains and discrete Dubrovin valuation rings.
• Extensions and deformations: The approach is pertinent in non-commutative algebraic geometry and module theory, particularly in understanding factorization, extension, and deformation properties under non-commutative operations.

6. Open Problems and Future Directions

The development of the star operation method in non-commutative settings opens several lines of inquiry:

• Left-right symmetry: It remains unresolved whether the right Prüfer ν-multiplication property implies a corresponding left property under general conditions, in contrast with known results for classical Prüfer orders in semi-hereditary contexts.
• Finite type variants: Further exploration of the interaction between the v-operation and its finite type variant, the T-operation, could clarify class group structures and stably presentable subclasses.
• Further generalizations: Investigation into whether specific non-commutative rings (e.g., skew group rings, Ore extensions, rings satisfying polynomial identities) conform to the Prüfer ν-multiplication property or admit similar star operation characterizations.
• Bridging commutative and non-commutative theories: The method may serve as a bridge, extending powerful commutative techniques and invariants into more general non-commutative contexts for ideal-theoretic and module-theoretic analysis.

7. Summary of Principal Formulas and Structural Insights

Concept	Formula/Description	Context
Divisorial closure ($v$-operation)	$I_v = (S : (S : I)_l)$	Key star operation in non-commutative rings
Prüfer ν-mult. invertibility (right)	$((S : I)_v\; I)' = S$; $(I (S : I)_v)' = O_1(I)$	Non-commutative generalization of invertibility
Overring inheritance	$((T : I') I')' = T$; $(I'(T : I')') = O_1(I')$	Prüfer property is overring-stable

These formulas encapsulate the translation of commutative multiplicative ideal theory into the non-commutative setting and yield new classification and invariance principles for orders in simple Artinian rings.

In conclusion, the star operation method, as manifested through the v-operation and the structure of right Prüfer ν-multiplication orders, equips non-commutative ideal theory with powerful techniques for analyzing divisorial closure, invertibility, and extension properties. It unifies disparate classes of orders, clarifies stability under overrings, and raises compelling open problems about the nature of multiplicative structures beyond commutative frameworks (1108.5448).