Ring (mathematics)
Algebraic structure with two binary operations, generalizing integer arithmetic.
A ring is an algebraic structure in mathematics consisting of a set equipped with two binary operations, typically called addition and multiplication, which behave similarly to integer addition and multiplication except that multiplication need not be commutative. Rings appear in a chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields. The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Richard Dedekind, David Hilbert, Abraham Fraenkel, and Emmy Noether.
- field
- Mathematics
- known_for
- Algebraic structure with addition and multiplication; foundation of commutative algebra, algebraic number theory, and algebraic geometry
- key_contributors
- Richard Dedekind, David Hilbert, Abraham Fraenkel, Emmy Noether
Lore & Background
A ring is a set R with two binary operations, addition (+) and multiplication (⋅), satisfying three sets of axioms: R is an abelian group under addition (associative, commutative, has additive identity 0, and additive inverses); R is a monoid under multiplication (associative, has multiplicative identity 1); and multiplication distributes over addition (both left and right). Some authors apply the term 'ring' to a structure without a multiplicative identity, called a 'rng' (missing 'i'). For example, the set of even integers with usual addition and multiplication is a rng but not a ring.
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.
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