division ring

division ring
Name	Division ring
Field	Algebra
Notable	William Rowan Hamilton, Ferdinand Georg Frobenius, Joseph Wedderburn, Skolem–Noether theorem

division ring

A division ring is a noncommutative generalization of a field: a set with two operations in which every nonzero element has a multiplicative inverse and addition forms an abelian group. It arose in the work of William Rowan Hamilton on quaternions and in classification results by Ferdinand Georg Frobenius and Joseph Wedderburn. Division rings serve as central objects in the theories of ring theory, module theory, algebraic number theory, and functional analysis.

Definition and basic properties

A division ring is a ring with unity in which every nonzero element is invertible under multiplication; left and right inverses coincide by associativity. Key algebraic consequences include that nonzero elements form a group under multiplication, the center is a field often called the center of the division ring, and finite division rings are fields by Wedderburn's little theorem proven by Joseph Wedderburn. The notion is tightly connected to central simple algebras studied by Richard Brauer and to normed division algebras classified by results of Ferdinand Georg Frobenius and later work of Adolf Hurwitz.

Examples and constructions

Classical examples include the real numbers ℝ and complex numbers ℂ as fields, and the noncommutative quaternion algebra discovered by William Rowan Hamilton. Matrix algebras over fields produce simple rings but are not division rings except in the 1×1 case; however, division algebras arise as skew fields such as the ring of quaternions and certain cyclic algebras constructed by methods of Emil Artin and Helmut Hasse. Local and global fields such as ℚ_p provide contexts for central division algebras via the Brauer group of a field studied by Richard Brauer and Helmut Hasse. Explicit finite-dimensional division algebras over number fields appear in the theory of quaternion algebras connected to Hilbert symbol computations and arithmetic of Shimura varieties.

Algebraic structure and subrings

Subrings and intermediate structures in division rings include the center (a field) and maximal subfields, which are fields whose degree equals the algebra dimension over the center; their existence is a topic in the theory of central simple algebras treated by Emil Artin and Richard Brauer. The study of automorphisms ties to Skolem–Noether theorem and to Galois cohomology developed by Jean-Pierre Serre and Alexander Grothendieck. Important constraints include Wedderburn's theorem for finite rings and various results on polynomial identities investigated by I. N. Herstein and Nathan Jacobson.

Modules and linear algebra over division rings

Vector spaces over a division ring generalize linear algebra; modules over a division ring are free and admit bases, leading to a well-defined dimension (also called rank) by arguments analogous to those of Ernst Steinitz. Linear transformations form matrix rings over the division ring, and the theory of determinants and characteristic polynomials requires adaptations; results such as the rational canonical form and Jordan form have analogues studied by Claude Chevalley and Irving Kaplansky. Projective geometry over division rings connects to work of H. S. M. Coxeter and the foundations of incidence geometry examined by Alfred North Whitehead and others.

Classification and Skolem–Noether theorem

Finite-dimensional central division algebras over a field are classified up to Brauer equivalence by the Brauer group introduced by Richard Brauer; local and global class field theory contributions by Helmut Hasse and John Tate provide finer invariants. The Skolem–Noether theorem, attributed to Thoralf Skolem and Max Noether, asserts that automorphisms of simple subalgebras inside central simple algebras are inner, constraining embeddings of division subrings and playing a central role in the structure theory developed by Nathan Jacobson and Emil Artin.

Applications and connections to other areas

Division rings play roles in many areas: noncommutative geometry studied by Alain Connes invokes division algebras in examples; representation theory of groups uses division algebras in endomorphism rings per Frobenius reciprocity contexts and the work of George Mackey; algebraic topology and K-theory utilize central simple algebras in computations pioneered by Michael Atiyah and Friedhelm Waldhausen. In number theory, quaternion algebras appear in the theory of modular forms and arithmetic geometry related to John Milnor and Andrew Wiles. Functional analytic settings produce C*-algebraic analogues where division-like behavior informs classification results by George Elliott.

Category:Algebra