Ideal theory grew up and renamed itself commutative algebra
Commutative algebra studies rings where multiplication swaps freely, plus their ideals and modules. Algebraic geometry and number theory both lean on it—so much that once-geometric words like dimension and localization now name purely algebraic ideas as well.
The subject was first called ideal theory. Typical commutative rings include polynomial rings and rings of algebraic integers, ordinary integers included. Noncommutative algebra covers rings that need not commute, along with representation theory and Banach algebras. In geometry, an affine variety lines up with a prime ideal inside a polynomial ring, and its points line up with the maximal ideals that contain that prime. The Zariski topology extends to the set of all prime ideals of any commutative ring: closed sets are primes containing a fixed ideal. The spectrum—primes with Zariski topology and localized sections—is the seed of Grothendieck's scheme theory, which then pushed commutative algebra further.
Richard Dedekind's ideals, building on Kummer and Kronecker, opened the story. David Hilbert coined "ring" to generalize number rings and pushed abstract methods over classical computation. Emmy Noether reframed results around the ascending chain condition—the Noetherian property. Hilbert's student Emanuel Lasker introduced primary ideals and a first form of the Lasker–Noether theorem. Wolfgang Krull made the field mature: localization, completion, regular local rings, and Krull dimension. His principal ideal theorem is often called the subject's most foundational theorem.
Modern emphasis falls on modules, which encompass both ideals and algebras over a ring—an approach credited mainly to Krull and Noether. A Noetherian ring is one in which every ideal is finitely generated; fields, the integers, and polynomial rings over them are Noetherian, the last by Hilbert's basis theorem. Quotients, localizations, and completions inherit the property. Many deep theorems—Lasker–Noether, Krull intersection, Nakayama—need Noetherian hypotheses. Primary decomposition generalizes unique factorization in the integers, and the radicals of the primary pieces are uniquely determined. Geometry and arithmetic keep feeding the same algebraic engine.
Source: Commutative algebra
