Commutative algebra

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A 1915 postcard from one of the pioneers of commutative algebra, Emmy Noether, to E. Fischer, discussing her work in commutative algebra Emmy noether postcard 1915.jpg
A 1915 postcard from one of the pioneers of commutative algebra, Emmy Noether, to E. Fischer, discussing her work in 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. [1]

Contents

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 (the main class of commutative rings occurring in algebraic number theory), 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 R 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 xyQ, either xQ or ynQ for some positive integer n. In Z, the primary ideals are precisely the ideals of the form (pe) where p is prime and e is a positive integer. Thus, a primary decomposition of (n) corresponds to representing (n) 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:

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 ij. Furthermore, if:

is decomposition of I with Rad(Pi) ≠ Rad(Pj) for ij, and both decompositions of I are irredundant (meaning that no proper subset of either {Q1, ..., Qt} or {P1, ..., Pk} yields an intersection equal to I), t = k and (after possibly renumbering the Qi) Rad(Qi) = Rad(Pi) for all i.

For any primary decomposition of I, the set of all radicals, that is, the set {Rad(Q1), ..., Rad(Qt)} remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the assassinator of the module R/I; that is, the set of all annihilators of R/I (viewed as a module over R) 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 fractions

.

where 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 (the set of prime ideals). [2] In this formulation, the Zariski-closed sets are taken to be the sets

where 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 (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. Thus, V(S) 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.

Connections with algebraic geometry

Commutative algebra (in the form of polynomial rings and their quotients, used in the definition of algebraic varieties) has always been a part of algebraic geometry. However, in the late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme. Their local objects are affine schemes or prime spectra, which are locally ringed spaces, which form a category that is antiequivalent (dual) to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced k-algebras. The gluing is along the Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set-theoretic sense is then replaced by a Zariski topology in the sense of Grothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc. Nowadays some other examples have become prominent, including the Nisnevich topology. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions, leading to Artin stacks and, even finer, Deligne–Mumford stacks, both often called algebraic stacks.

See also

Notes

  1. Atiyah and Macdonald, 1969, Chapter 1
  2. Dummit, D. S.; Foote, R. (2004). Abstract Algebra (3 ed.). Wiley. pp.  71–72. ISBN   9780471433347.

Related Research Articles

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left ideals has a largest element; that is, there exists an n such that:

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.

<span class="mw-page-title-main">Emmy Noether</span> German mathematician (1882–1935)

Amalie Emmy Noether was a German mathematician who made many important contributions to abstract algebra. She proved Noether's first and second theorems, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.

<span class="mw-page-title-main">Zariski topology</span> Topology on prime ideals and algebraic varieties

In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space.

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be any Noetherian local ring with unique maximal ideal m, and suppose a1, ..., an is a minimal set of generators of m. Then Krull's principal ideal theorem implies that n ≥ dim A, and A is regular whenever n = dim A.

In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.

In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals. The theorem was first proven by Emanuel Lasker for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by Emmy Noether.

In ring theory and homological algebra, the global dimension of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings. By a theorem of Jean-Pierre Serre, global dimension can be used to characterize within the class of commutative Noetherian local rings those rings which are regular. Their global dimension coincides with the Krull dimension, whose definition is module-theoretic.

In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals.

In abstract algebra, 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 a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions R on a space X concentrates on a formal neighborhood of a point of X: heuristically, this is a neighborhood so small that all Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in the case when R has a metric given by a non-Archimedean absolute value.

In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over A.

This is a glossary of commutative algebra.

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