Commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and p-adic integers.
Commutative algebra is the main technical tool of algebraic geometry, and many results and concepts of commutative algebra are strongly related with geometrical concepts.
The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras.
Overview
Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry.Several concepts of commutative algebras have been developed in relation with algebraic number theory, such as Dedekind rings, integral extensions, and valuation rings.
Polynomial rings in several indeterminates over a field are examples of commutative rings. Since algebraic geometry is fundamentally the study of the common zeros of these rings, many results and concepts of algebraic geometry have counterparts in commutative algebra, and their names recall often their geometric origin; for example "Krull dimension", "localization of a ring", "local ring", "regular ring".
An affine algebraic variety corresponds to a prime ideal in a polynomial ring, and the points of such an affine variety correspond to the maximal ideals that contain this prime ideal. The Zariski topology, originally defined on an algebraic variety, has been extended to the sets of the prime ideals of any commutative ring; for this topology, the closed sets are the sets of prime ideals that contain a given ideal.
The spectrum of a ring is a ringed space formed by the prime ideals equipped with the Zariski topology, and the localizations of the ring at the open sets of a basis of this topology. This is the starting point of scheme theory, a generalization of algebraic geometry introduced by Grothendieck, which is strongly based on commutative algebra, and has induced, in turns, many developments of commutative algebra.
History
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. 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 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. 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 rings and Krull rings. 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 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 Kronecker's work, the modern approach to commutative algebra using module theory is usually credited to Krull and Noether.
Main tools and results
Noetherian rings
A Noetherian ring, named after Emmy Noether, is a ring in which every ideal is finitely generated; that is, all elements of any ideal can be written as a linear combinations of a finite set of elements, with coefficients in the ring.Many commonly considered commutative rings are Noetherian, in particular, every field, the ring of the integer, and every polynomial ring in one or several indeterminates over them. The fact that polynomial rings over a field are Noetherian is called Hilbert's basis theorem.
Moreover, many ring constructions preserve the Noetherian property. In particular, if a commutative ring is Noetherian, the same is true for every polynomial ring over it, and for every quotient ring, localization, or completion of the ring.
The importance of the Noetherian property lies in its ubiquity and also in the fact that many important theorems of commutative algebra require that the involved rings are Noetherian, This is the case, in particular of Lasker–Noether theorem, the Krull intersection theorem, and Nakayama's lemma.
Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on prime ideals, which implies that every Noetherian local ring has a finite Krull dimension.
Primary decomposition
An ideal Q of a ring is said to be primary if Q is proper and whenever xy ∈ Q, either x ∈ Q or yn ∈ Q for some positive integer n. In Z, the primary ideals are precisely the ideals of the form where p is prime and e is a positive integer. Thus, a primary decomposition of corresponds to representing as the intersection of finitely many primary ideals.The Lasker–Noether theorem, given here, may be seen as a certain generalization of the fundamental theorem of arithmetic:
For any primary decomposition of I, the set of all radicals, that is, the set remains the same by the Lasker–Noether theorem. In fact, it turns out that the set is precisely the assassinator of the module R/''I; that is, the set of all annihilators of R''/I that are prime.
Localization
The localization is a formal way to introduce the "denominators" to a given ring or a module. That is, it introduces a new ring/module out of an existing one so that it consists of fractionswhere the denominators s range in a given subset S of R. The archetypal example is the construction of the ring Q of rational numbers from the ring Z of integers.
Completion
A completion is any of several related functors on rings and modules that result in complete topological rings and modules. 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.Zariski topology on prime ideals
The Zariski topology defines a topology on the spectrum of a ring. In this formulation, the Zariski-closed sets are taken to be the setswhere A is a fixed commutative ring and I is an ideal. This is defined in analogy with the classical Zariski topology, where closed sets in affine space are those defined by polynomial equations. To see the connection with the classical picture, note that for any set S of polynomials, it follows from Hilbert's Nullstellensatz that the points of V are exactly the tuples such that the ideal 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. Thus, V is "the same as" the maximal ideals containing S. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.