Commutative algebra

The subject, first known as ideal theory, began with Richard Dedekind's work on ideals, itself based on the earlier work of Ernst Kummer and Leopold Kronecker.

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.

Another important milestone was the work of Hilbert's student Emanuel Lasker, who introduced primary ideals 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 and completion of a ring, as well as that of regular local rings.

Though it was already incipient in Kronecker's work, the modern approach to commutative algebra using module theory is usually credited to Krull and Noether.

The fact that polynomial rings over a field are Noetherian is called Hilbert's basis theorem.

The Lasker–Noether theorem, given here, may be seen as a certain generalization of the fundamental theorem of arithmetic: Lasker-Noether Theorem — Let R be a commutative Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals; that is: with Qi primary for all i and Rad(Qi) ≠ Rad(Qj) for i ≠ j.

Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings.

Complete commutative rings have simpler structure than the general ones and Hensel's lemma applies to them.

To see the connection with the classical picture, note that for any set S of polynomials (over an algebraically closed field), it follows from Hilbert's Nullstellensatz that the points of V(S) (in the old sense) are exactly the tuples (a1, ..., an) such that the ideal (x1 - a1, ..., xn - an) contains S; moreover, these are maximal ideals and by the "weak" Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form.

However, in the late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme.