{
  "id": "ring",
  "domain": "algebra",
  "type": "Concept",
  "title": "Ring",
  "slug": "ring",
  "url": "/mathematics/algebra/ring/",
  "summary": "A set with addition and multiplication where addition is an abelian group, multiplication is associative, and distributive laws hold. Generalises the integers; includes polynomials, matrices, and modular arithmetic.",
  "formal_definition": "(R, +, ·) is a ring if (R, +) is an abelian group, multiplication is associative, and both distributive laws hold.",
  "key_results": [
    { "result": "Hilbert Basis Theorem", "statement": "The polynomial ring R[x] is Noetherian if R is Noetherian; every ideal in a polynomial ring over ℤ is finitely generated." },
    { "result": "Unique factorisation failure", "statement": "In ℤ[√−5], 6 = 2·3 = (1+√−5)(1−√−5) — two distinct factorisations. Motivated Dedekind's ideal theory." }
  ],
  "introduced_by": "Term 'ring' (Zahlring) — David Hilbert, Zahlbericht, 1897. Abstract definition — Emmy Noether, 1921.",
  "added_to_codex": "2026-05-27",
  "last_verified": "2026-05-27",
  "sources": [
    { "tier": 1, "citation": "Dummit, D.S. and Foote, R.M. (2004). Abstract Algebra. 3rd ed. John Wiley & Sons.", "chapters": "7–9" },
    { "tier": 1, "citation": "Atiyah, M.F. and Macdonald, I.G. (1969). Introduction to Commutative Algebra. Addison-Wesley." },
    { "tier": 2, "citation": "Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer." },
    { "tier": 2, "citation": "Noether, E. (1921). Idealtheorie in Ringbereichen. Mathematische Annalen, 83, 24–66.", "note": "Primary source" }
  ],
  "relationships": [
    { "type": "generalises", "target": "group", "label": "Group" },
    { "type": "specialises_to", "target": "field", "label": "Field" },
    { "type": "key_person", "target": "noether-emmy", "label": "Emmy Noether" },
    { "type": "key_person", "target": "dedekind-richard", "label": "Richard Dedekind" }
  ],
  "entry_status": "live",
  "reading_levels_available": ["curious", "exploring", "deep_dive"],
  "completeness": 1.0,
  "codex_version": "1.1"
}