Ring (mathematics)

Definition and key properties

Formally, a ring (R, +, ·) satisfies these conditions: the set R is closed under + and ·; (R, +) is an abelian group; multiplication is associative; and multiplication distributes over addition on both sides. Some rings also have a multiplicative identity (1), and some have commutative multiplication. Important derived concepts include units (invertible elements), zero divisors, and ideals. For axioms and examples of variants see axioms and variants.

Common examples and constructions

• The integers Z, rational numbers Q, real numbers R and complex numbers C (all commutative rings, Q, R, C are fields).
• Rings of polynomials R[x], used to model algebraic expressions and roots.
• Matrix rings M_n(R), typically noncommutative when n>1.
• Quotient rings such as Z/nZ and rings of functions on a space.

See introductions to polynomial and matrix examples at polynomial rings and matrix rings.

History and development

The modern term "ring" comes from the German Zahlring used by David Hilbert in the late 19th century to describe sets of algebraic integers. Later 20th-century algebraists, notably Emmy Noether, developed the abstract theory of rings and ideals; this axiomatic approach clarified structures that appear in algebraic number theory and algebraic geometry. For historical notes see Hilbert and Noether.

Variants, importance and applications

Distinguished classes include commutative rings, integral domains (no nonzero zero divisors), division rings and fields. Ideals and ring homomorphisms form the language for solving equations and constructing quotient rings; modules over a ring generalize vector spaces. Rings are central in algebraic geometry (coordinate rings of varieties), in number theory (rings of integers, modular arithmetic), and in coding theory and cryptography. Further resources: ideals, modules, and applications.