{"id":"ideal","domain":"algebra","type":"Concept","title":"Ideal","slug":"ideal","url":"/mathematics/algebra/ideal/","summary":"A subset of a ring that absorbs multiplication. Invented by Kummer (1840s) to rescue unique factorization; formalized by Dedekind. Enables quotient rings, prime/maximal ideals.","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."},{"tier":1,"citation":"Atiyah, M.F. and Macdonald, I.G. (1969). Introduction to Commutative Algebra."},{"tier":2,"citation":"Edwards, H.M. (1977). Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory."},{"tier":3,"citation":"Reid, M. (1988). Undergraduate Algebraic Geometry."}],"relationships":[{"type":"defined_within","target":"ring","label":"Ring"},{"type":"motivated_by","target":"fermats-last-theorem","label":"Fermat's Last Theorem"},{"type":"generalises","target":"prime-number","label":"Prime Number"},{"type":"analogous_construction","target":"homomorphism-isomorphism","label":"Homomorphism & Isomorphism — quotient groups"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.4"}