# Ring

**Type:** Concept  
**Domain:** Algebra  
**Codex URL:** /mathematics/algebra/ring/  
**Entry status:** Live — v1.0 (2026-05-27)

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## Curious Level

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. The integers ℤ are the prototypical ring. Polynomials, matrices, and modular arithmetic are all rings.

The term "ring" (Zahlring) was introduced by Hilbert in 1897. The abstract definition was established by Emmy Noether in her 1921 paper "Idealtheorie in Ringbereichen."

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## Exploring Level

### Definition

(R, +, ·) is a **ring** if:
1. (R, +) is an abelian group (additive identity 0; every a has −a)
2. Multiplication is associative: (a·b)·c = a·(b·c)
3. Distributive laws: a·(b+c) = ab+ac and (a+b)·c = ac+bc

A **commutative ring** additionally has a·b = b·a. A **unital ring** has a multiplicative identity 1. An **integral domain** is a commutative unital ring with no zero divisors.

### Hierarchy

Ring ⊇ Commutative ring ⊇ Integral domain ⊇ UFD ⊇ PID ⊇ Euclidean domain ⊇ Field

ℤ is a PID but not a field. ℤ[x] is a UFD but not a PID. ℤ[i] is a Euclidean domain.

### Ideals

An ideal I ⊆ R satisfies r·i ∈ I and i·r ∈ I for all r ∈ R, i ∈ I. Every ideal in ℤ is of the form nℤ. Ideals are kernels of ring homomorphisms and allow construction of quotient rings R/I.

### Unique factorisation failure

In ℤ[√−5]: 6 = 2·3 = (1+√−5)(1−√−5), two distinct factorisations. This motivated Dedekind's ideal theory (1871).

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## Deep Dive

### Historical development

Gauss (1832): Gaussian integers ℤ[i]. Kummer (1844): "ideal numbers" for cyclotomic integers. Dedekind (1871): ideals as subsets of rings. Hilbert (1897): term "Zahlring," Hilbert Basis Theorem. Noether (1921): abstract ring theory, ascending chain condition, Noetherian rings.

### Emmy Noether

Noether's 1921 paper established Noetherian rings (ACC condition), proved ideals in Noetherian rings are finitely generated. Noetherian rings are the central objects of commutative algebra and algebraic geometry.

### Localization

S⁻¹R formally adjoins inverses of elements of S ⊆ R. Algebraic analogue of restricting to a neighbourhood; fundamental to scheme theory (Grothendieck).

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## Sources

### Tier 1
- Dummit, D.S. and Foote, R.M. (2004). *Abstract Algebra*. 3rd ed. John Wiley & Sons. [Chapters 7–9]
- Atiyah, M.F. and Macdonald, I.G. (1969). *Introduction to Commutative Algebra*. Addison-Wesley.

### Tier 2
- Eisenbud, D. (1995). *Commutative Algebra with a View Toward Algebraic Geometry*. Springer.
- Noether, E. (1921). Idealtheorie in Ringbereichen. *Mathematische Annalen*, 83, 24–66. [Primary source]

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*