ring (mathematics)

ring (mathematics)
Name	Ring (mathematics)
Type	Algebraic structure
Axioms	Additive abelian group, multiplicative semigroup, distributivity

ring (mathematics) is an algebraic structure consisting of a set equipped with two binary operations called addition and multiplication, satisfying axioms that generalize the arithmetic of integers and the algebra of polynomials. Rings arise across modern mathematics, connecting Isaac Newton, Carl Friedrich Gauss, David Hilbert, Emmy Noether, Alexander Grothendieck, and Emil Artin through the development of number theory, algebraic geometry, and homological algebra. They provide foundational language for topics studied by researchers at institutions such as University of Göttingen, University of Cambridge, Princeton University, Harvard University, and Massachusetts Institute of Technology.

Definition and basic examples

A ring is defined by axioms formalized in the work of Richard Dedekind, Ernst Steinitz, and Emmy Noether: the underlying set is an abelian group under addition and a semigroup under multiplication, with multiplication distributive over addition. Standard examples include the ring of integers ℤ studied by Leonhard Euler and Pierre de Fermat, polynomial rings like K[x] used by Évariste Galois and Niels Henrik Abel, matrix rings M_n(R) prominent in work by Arthur Cayley and William Rowan Hamilton, and rings of continuous functions considered by Bernhard Riemann. Other canonical examples are group rings arising in the research of William Burnside and Emil Artin, formal power series rings used in Oscar Zariski's studies, and rings of algebraic integers central to Andrew Wiles's proof of the Taniyama–Shimura conjecture.

Ring homomorphisms and ideals

Morphisms between rings, or ring homomorphisms, generalize maps like the inclusion ℤ → ℚ and the evaluation maps used by Niels Abel and Joseph-Louis Lagrange. Ideals, introduced by Richard Dedekind and systematized by Emmy Noether, serve as kernels of homomorphisms and as building blocks for quotient rings R/I, paralleling constructions used in Algebraic Number Theory by Kurt Heegner and Carl Friedrich Gauss. Prime ideals and maximal ideals mirror primes in ℤ and play a central role in the work of Oscar Zariski and Alexander Grothendieck on spectra of rings, while radical ideals appear in the Nullstellensatz from David Hilbert.

Ring-theoretic constructions and classifications

Standard constructions include direct sums and products studied by Felix Klein, localizations developed by David Hilbert and applied by Jean-Pierre Serre, completions used by Kurt Hensel and Alexander Grothendieck, and tensor products central to Samuel Eilenberg and Saunders Mac Lane's category theory. Classification results, such as the structure theorem for finitely generated modules over principal ideal domains explored by Emmy Noether and Emil Artin, and the Artin–Wedderburn theorem named after Emil Artin and Joseph Wedderburn, organize semisimple and simple rings as matrix algebras over division rings. The development of noncommutative ring theory owes much to Richard Brauer, Israel Gelfand, and Israel Gelfand's collaborators, influencing the study of C*-algebras by John von Neumann and Alfred Tarski.

Modules and representations over rings

Modules generalize vector spaces over fields and were formalized in the categorical framework advanced by Saunders Mac Lane and Samuel Eilenberg. Module theory underpins representation theory of groups as modules over group rings, an approach pioneered by Frobenius and William Burnside, and expanded in the work of Issai Schur, George Mackey, and Roger Howe. Projective, injective, and flat modules, central to homological methods developed by Henri Cartan and Jean-Pierre Serre, facilitate resolutions and derived functors used throughout Algebraic Topology and Algebraic Geometry by researchers at Institute for Advanced Study and Institut des Hautes Études Scientifiques.

Special classes of rings

Commutative rings with identity include principal ideal domains, unique factorization domains studied by Gauss and David Hilbert, Dedekind domains investigated by Richard Dedekind, and Noetherian rings named for Emmy Noether. Noncommutative examples encompass division rings treated by Joseph Wedderburn, simple rings classified by Emil Artin, and von Neumann regular rings examined by John von Neumann. Topological and analytic flavors appear in Banach algebras associated with Stefan Banach and C*-algebras in the work of John von Neumann and Gelfand. Graded rings and superalgebras influence algebraic geometry and theoretical physics through contributions by Alexander Grothendieck, Edward Witten, and Michael Atiyah.

Applications and connections to other areas

Rings interface with algebraic number theory in the study of algebraic integers central to Andrew Wiles and Gerhard Frey's work, with algebraic geometry via schemes developed by Alexander Grothendieck and Michel Demazure, with representation theory through the Langlands program promoted by Robert Langlands, and with topology through cohomology rings studied by Henri Cartan and Jean Leray. Computational aspects link to computer algebra systems and algorithms influenced by François Lecomte and Donald Knuth, while applications in mathematical physics connect rings to quantum theory via operator algebras used by Werner Heisenberg and Paul Dirac.

Category:Algebra