Ore condition

Ore condition. In the branch of abstract algebra known as ring theory, the Ore condition is a crucial technical criterion that allows for the construction of a division ring of fractions from certain noncommutative rings, generalizing the familiar process of forming the field of fractions from an integral domain. First introduced by the Norwegian mathematician Øystein Ore in a seminal 1931 paper, this condition ensures that a ring can be embedded into a skew field where every non-zero element has a multiplicative inverse. The study of this condition is central to understanding the structure of noncommutative algebra and has profound implications in areas like representation theory and algebraic geometry.

Definition and motivation

The classical construction of the field of fractions for an integral domain, such as forming the rational numbers from the integers, relies heavily on the commutative property of multiplication. For a general noncommutative ring \(R\) with no zero divisors, this process fails because one cannot simply equate fractions \(a s^{-1}\) and \(t^{-1} b\) without a method to "common denominators." The right Ore condition provides this method: it requires that for any elements \(a \in R\) and a regular element \(s \in R\) (an element that is not a zero divisor), there exist elements \(b \in R\) and a regular \(t \in R\) such that \(a t = s b\). This equation allows fractions to be rewritten with a common denominator on the right, enabling a consistent definition of addition. The analogous left Ore condition is defined symmetrically. A ring satisfying both is called an Ore domain, and the resulting ring of fractions is a division ring, as established in Ore's theorem.

Examples and non-examples

A fundamental example is any commutative integral domain, such as the ring of polynomials \(\mathbb{Z}[x]\), which trivially satisfies the condition. Important noncommutative examples include principal ideal domains like the first Weyl algebra \(A_1(k)\) over a field \(k\) of characteristic zero, and more generally, many Noetherian domains such as universal enveloping algebras of finite-dimensional Lie algebras. The condition also holds for group rings \(K[G]\) where \(G\) is a polycyclic-by-finite group and \(K\) is a field, a result connected to the work of Donald S. Passman. A classic non-example is the free algebra \(K\langle x, y \rangle\) on two generators over a field \(K\); it contains no non-trivial zero divisors but fails the Ore condition, as the right ideals \(xR\) and \(yR\) intersect trivially, preventing the formation of a division ring of fractions.

Properties and consequences

The primary consequence of the Ore condition is the existence of the Ore localization, a universal construction yielding a division ring \(Q\) and an injective homomorphism from the original ring into \(Q\) where every regular element becomes a unit. This localization preserves many ring-theoretic properties; for instance, if the original ring is a Noetherian domain, then its Ore localization is a simple Artinian ring, a fact underpinning Goldie's theorem. The condition is also intimately linked to the structure of projective modules and homological dimension; for an Ore domain, all finitely generated torsion-free modules embed into free modules. Furthermore, the Jacobson radical of the Ore localization relates to that of the original ring, and the condition ensures that the ring of quotients has a well-defined rank of a module.

Generalizations and related concepts

The Ore condition has been generalized in several significant directions. One major extension is Malcev–Neumann construction, which can form division rings from certain crossed products even when the Ore condition fails. The concept of a right reversible semigroup is the semigroup-theoretic analogue. In the context of module theory, the condition is generalized by the notion of a right denominator set, which allows localization with respect to multiplicative sets that may contain zero divisors, leading to the theory of rings of quotients developed by Lance W. Small and others. The related common multiple property is essential in the study of noncommutative unique factorization domains. Furthermore, the work of Paul Moritz Cohn on firs and semifirs explores rings where the Ore condition holds for finitely generated right ideals, connecting to the syzygy theorem and homological algebra. Category:Abstract algebra Category:Ring theory Category:Mathematical theorems