Ring (mathematics)

Lore & Background

A ring is a set equipped with two binary operations, typically called addition and multiplication, that behave similarly to the addition and multiplication of integers, except that multiplication need not be commutative. The elements of a ring can be numbers, such as integers or complex numbers, or non-numerical objects like polynomials, square matrices, functions, and power series. Formally, a ring is an abelian group under addition (meaning addition is associative, commutative, has an additive identity element called 0, and every element has an additive inverse) and a monoid under multiplication (multiplication is associative and has a multiplicative identity element called 1). Multiplication must also be distributive over addition, both on the left and right. Some authors use the term "ring" for a structure that omits the requirement for a multiplicative identity, calling such a structure a "rng" (a ring missing the letter 'i'). For example, the set of even integers with usual addition and multiplication forms a rng but not a ring. A commutative ring is one where multiplication is commutative, a property with profound implications; the theory of commutative rings, known as commutative algebra, is a major branch of ring theory deeply influenced by algebraic number theory and algebraic geometry. Examples of commutative rings include every field (such as the real or complex numbers), the integers, polynomial rings, and the coordinate ring of an affine algebraic variety. Noncommutative rings include rings of square matrices, group rings in representation theory, operator algebras, rings of differential operators, and cohomology rings in topology. The conceptualization of rings developed from the 1870s to the 1920s, with key contributions from Richard Dedekind, David Hilbert, Abraham Fraenkel, and Emmy Noether, initially as a generalization of Dedekind domains in number theory and polynomial rings in algebraic geometry.

Reader's Guide

Rings are fundamental in modern mathematics. They were first formalized as a generalization of Dedekind domains in number theory and of polynomial rings and rings of invariants in algebraic geometry and invariant theory. Commutative rings—where multiplication is commutative—are the subject of commutative algebra, a major branch of ring theory deeply influenced by algebraic number theory and algebraic geometry. Examples of commutative rings include every field (e.g., real or complex numbers), the integers, polynomials with coefficients in another ring, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Noncommutative rings include n×n real square matrices (n≥2), group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology. Rings have proven useful in geometry and analysis as well.

Did You Know?

• Ring multiplication is not required to be commutative; a ring with commutative multiplication is called a commutative ring.
• The set of even integers with usual addition and multiplication is a rng (lacking multiplicative identity), not a ring.
• If 0 = 1 in a ring, the ring has only one element and is called the zero ring.
• The additive group of a ring is abelian, though this can be inferred from the other ring axioms only when a multiplicative identity is present.