A ring is a set with two operations — addition and multiplication — where addition forms an abelian group, multiplication is associative, and multiplication distributes over addition. Rings generalise the integers, and include polynomials, matrices, and modular arithmetic as natural examples.
What is a ring?
The integers ℤ = {…, −2, −1, 0, 1, 2, …} have two natural operations: you can add them and you can multiply them. Addition makes the integers into a group (you can undo addition by subtracting). Multiplication is associative and distributes over addition — a × (b + c) = ab + ac. But you generally can't undo multiplication (6 ÷ 4 is not an integer).
A ring is the abstract pattern extracted from this: a set with addition and multiplication where addition is "nice" (forms an abelian group) and multiplication is associative and plays well with addition (distributive law).
Examples include:
The integers ℤ — the prototypical ring
The integers mod n — clock arithmetic (ℤ/nℤ)
Polynomials with integer coefficients ℤ[x]
2×2 matrices over the reals M₂(ℝ)
Even numbers 2ℤ — a ring without a multiplicative identity
Rings are everywhere in mathematics. Every number system you're likely to encounter is either a ring or built from rings. The study of rings is called ring theory, and it evolved in the 19th century from attempts to generalise number theory and understand factorisation.
Formal definition and key properties
Definition
A ring is a triple (R, +, ·) where R is a set and +, · are binary operations satisfying:
(R, +) is an abelian group (identity: 0, inverse of a: −a)
Multiplication is associative: (a·b)·c = a·(b·c) for all a, b, c ∈ R
A ring with a multiplicative identity (a 1 such that 1·a = a·1 = a for all a) is a unital ring. Most modern texts require a 1. A ring where multiplication is also commutative (a·b = b·a) is a commutative ring. A commutative unital ring with no zero divisors is an integral domain.
Key examples
Ring
Commutative?
Has 1?
Domain?
ℤ (integers)
Yes
Yes
Yes
ℤ/nℤ (integers mod n)
Yes
Yes
iff n prime
ℤ[x] (integer polynomials)
Yes
Yes
Yes
M₂(ℝ) (2×2 real matrices)
No
Yes
No
2ℤ (even integers)
Yes
No
Yes
Ideals and quotient rings
An ideal I ⊆ R is a subring closed under multiplication by all ring elements: r·i ∈ I and i·r ∈ I for all r ∈ R, i ∈ I. Ideals play the role for rings that normal subgroups play for groups: they are precisely the kernels of ring homomorphisms, and they allow the construction of quotient rings R/I.
In the integers, every ideal is of the form nℤ = {…, −2n, −n, 0, n, 2n, …} for some n ∈ ℤ. The quotient ℤ/nℤ is the ring of integers modulo n.
Factorisation and unique factorisation domains
In ℤ, every integer factors uniquely into primes. Rings that share this property are called unique factorisation domains (UFDs). Not all rings have unique factorisation: in ℤ[√−5], the number 6 factors as both 2·3 and (1+√−5)(1−√−5), with none of these factorisable further. This failure of unique factorisation in rings of algebraic integers motivated Kummer's introduction of "ideal numbers" (1844) and eventually the modern theory of ideals (Dedekind, 1871).
Structure theorems, classification, and historical roots
Historical development
The concept of a ring emerged from 19th-century number theory. Gauss (1832) introduced Gaussian integers ℤ[i] to study sums of two squares. Kummer (1844) introduced "ideal numbers" to restore unique factorisation in rings of cyclotomic integers, motivated by attempts to prove Fermat's Last Theorem. Dedekind (1871) reformulated Kummer's ideal numbers as subsets of rings — the modern concept of ideals. Hilbert (1897) axiomatised the theory. The term "ring" (Zahlring) was introduced by Hilbert (1897) in his Zahlbericht. The abstract definition was consolidated by Emmy Noether and Emil Artin in the 1920s.
Emmy Noether and abstract ring theory
Emmy Noether's 1921 paper "Idealtheorie in Ringbereichen" established the modern abstract theory of rings. She introduced the ascending chain condition (ACC) — now called the Noetherian condition — and proved that in a Noetherian ring, every ideal is finitely generated. Noetherian rings are the central objects of commutative algebra and algebraic geometry. The Hilbert Basis Theorem (1888) states that ℤ[x₁, …, xₙ] is Noetherian — every ideal in a polynomial ring over a Noetherian ring is finitely generated.
The hierarchy of commutative rings
Ring theory organises commutative rings into a hierarchy of increasingly restrictive conditions:
Ring ⊇ Commutative ring ⊇ Integral domain (no zero divisors) ⊇ Noetherian domain ⊇ UFD (unique factorisation) ⊇ PID (every ideal principal) ⊇ Euclidean domain (Euclidean algorithm applies) ⊇ Field (every non-zero element has a multiplicative inverse)
The integers ℤ are a PID but not a field. The polynomial ring ℤ[x] is a UFD but not a PID (the ideal (2, x) is not principal). The Gaussian integers ℤ[i] are a Euclidean domain with norm N(a+bi) = a² + b².
Localization and algebraic geometry
The localization of a ring R at a multiplicative set S — written S⁻¹R — formally adjoins inverses of elements of S. Localising ℤ at the set of integers coprime to p gives ℤ_(p), the integers localised at p, in which every fraction with denominator not divisible by p is an element. Localization is the algebraic analogue of restricting to a neighbourhood in topology, and is fundamental to scheme theory and modern algebraic geometry (Grothendieck).
Sources
Dummit, D.S. and Foote, R.M. (2004). Abstract Algebra. 3rd ed. John Wiley & Sons. — Chapters 7–9 cover ring theory thoroughly. Tier 1
Atiyah, M.F. and Macdonald, I.G. (1969). Introduction to Commutative Algebra. Addison-Wesley. — Concise, authoritative reference for commutative ring theory. Tier 1
Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer. — Comprehensive graduate treatment. Tier 2
Noether, E. (1921). Idealtheorie in Ringbereichen. Mathematische Annalen, 83, 24–66. — Original paper establishing modern abstract ring theory. Tier 2 (primary source)
Relationships
GeneralisesGroup — additive structure of a ring is an abelian group
Specialised toField — a field is a commutative ring where every non-zero element has a multiplicative inverse
Key person Emmy Noether (1882–1935) — abstract ring theory, Noetherian rings
Key person Richard Dedekind (1831–1916) — ideals, algebraic number theory