Editing Commutative algebra (section)

== History ==
The subject, first known as [[ideal theory]], began with [[Richard Dedekind]]'s work on [[Ideal (ring theory)|ideal]]s, itself based on the earlier work of [[Ernst Kummer]] and [[Leopold Kronecker]]. Later, [[David Hilbert]] introduced the term ''ring'' to generalize the earlier term ''number ring''. Hilbert introduced a more abstract approach to replace the more concrete and computationally oriented methods grounded in such things as [[complex analysis]] and classical [[invariant theory]].  In turn, Hilbert strongly influenced [[Emmy Noether]], who recast many earlier results in terms of an [[ascending chain condition]], now known as the Noetherian condition. Another important milestone was the work of Hilbert's student [[Emanuel Lasker]], who introduced [[primary ideal]]s and proved the first version of the [[Lasker–Noether theorem]].

The main figure responsible for the birth of commutative algebra as a mature subject was [[Wolfgang Krull]], who introduced the fundamental notions of [[Localization of a ring|localization]] and [[Completion (ring theory)|completion]] of a ring, as well as that of [[regular local ring]]s. He established the concept of the [[Krull dimension]] of a ring, first for [[Noetherian rings]] before moving on to expand his theory to cover general [[valuation ring]]s and [[Krull ring]]s. To this day, [[Krull's principal ideal theorem]] is widely considered the single most important foundational theorem in commutative algebra. These results paved the way for the introduction of commutative algebra into algebraic geometry, an idea which would revolutionize the latter subject.

Much of the modern development of commutative algebra emphasizes [[module (mathematics)|modules]].  Both  ideals of a ring ''R'' and ''R''-algebras are special cases of ''R''-modules, so module theory encompasses both ideal theory and the theory of [[ring extensions]].  Though it was already incipient in [[Leopold Kronecker|Kronecker's]] work, the modern approach to commutative algebra using module theory is usually credited to [[Wolfgang Krull|Krull]] and [[Emmy Noether|Noether]].