Ore domain

Ore domain. In the branch of abstract algebra known as ring theory, an Ore domain is a specific type of integral domain that permits the construction of a field of fractions in a manner analogous to the commutative case. The condition, introduced by Øystein Ore in the 1930s, ensures that the domain can be embedded into a division ring, providing a powerful tool for non-commutative analysis. This concept generalizes the familiar process of forming the field of rational numbers from the integers to a wide class of non-commutative rings.

Definition and basic properties

An integral domain \( R \) is called a right Ore domain if for any two nonzero elements \( a, b \in R \), there exist nonzero elements \( c, d \in R \) such that \( ad = bc \). This equality is known as the right Ore condition. The left Ore condition is defined symmetrically, and a domain satisfying both is simply termed an Ore domain. A fundamental theorem states that a domain is a right Ore domain if and only if it can be embedded in a division ring via a universal construction of right fractions. Key properties include that every right ideal of a right Ore domain contains a regular element, and the intersection of any two nonzero right ideals is nonzero. Important subclasses include right Bézout domains and right principal ideal domains, which automatically satisfy the Ore condition. The condition is also preserved under certain ring extensions and constructions.

Examples

The most classical example is the ring of integers \(\mathbb{Z}\), which is commutative and trivially an Ore domain, leading to the field of rational numbers. In the non-commutative setting, any right principal ideal domain, such as the first Weyl algebra over a field of characteristic zero, is an Ore domain. The skew polynomial ring \( K[x; \sigma] \) over a division ring \( K \) with an automorphism \( \sigma \) is also an Ore domain. Furthermore, every Noetherian domain that is an integral domain is an Ore domain, as established by results related to Goldie's theorem. Important examples in analysis include certain rings of differential operators, like sections of the sheaf of differential operators on a smooth manifold. Domains that are not Ore include free algebras over a field, such as the free algebra \( K\langle x, y \rangle \), where the condition fails.

Non-commutative analogues and generalizations

The Ore condition is central to the theory of non-commutative localization. A major generalization is the concept of a right Goldie ring, where Goldie's theorem states that a prime ring is a right order in a simple Artinian ring if and only if it satisfies the right Goldie condition. The Ore condition itself can be relaxed to the reversible Ore condition or studied in the context of semigroup rings and group rings. For semiprime rings, the notion of a classical ring of quotients is directly tied to the Ore property. The theory extends to module theory through the study of torsion theory and the Gabriel filter, where the Ore condition characterizes when a localizable prime ideal exists.

Relation to other ring-theoretic properties

Ore domains sit in a rich hierarchy of algebraic structures. Every right principal ideal domain and every right Bézout domain is a right Ore domain. Furthermore, every right Noetherian domain is an Ore domain, a consequence of the ascending chain condition on right ideals. The property is closely linked to the notion of a right order in a division ring, analogous to how \(\mathbb{Z}\) is an order in \(\mathbb{Q}\). However, an Ore domain need not be Noetherian; examples exist that are right Ore domains but not right Noetherian. The condition also interacts with chain conditions on annihilators and is preserved in polynomial ring extensions under certain conditions on the coefficient ring.

Ore condition and localization

The primary utility of the Ore condition is enabling the construction of a ring of fractions, often called the Ore localization. For a right Ore domain \( R \), one can form the division ring of right fractions \( Q_{cl}(R) \) by considering equivalence classes of pairs \((a, s)\) with \(s\) regular, mimicking the construction of \(\mathbb{Q}\) from \(\mathbb{Z}\). The process requires the Ore condition to ensure that fractions can be brought to a common denominator. This localization is a special case of the more general non-commutative localization theory developed by Paul Morita and others. It is fundamental in areas like non-commutative algebraic geometry, the study of quantum groups, and the analysis of enveloping algebras of Lie algebras, where the field of fractions provides a manageable ambient division ring.

Category:Ring theory Category:Algebraic structures