Commutative algebra, first known as
ideal theory
In mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only consid ...
, 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 integer
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s, including the ordinary
integers
; and
''p''-adic integers.
Commutative algebra is the main technical tool in the local study of
schemes.
The study of rings that are not necessarily commutative is known as
noncommutative algebra; it includes
ring theory
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 r ...
,
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, and the theory of
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
s.
Overview
Commutative algebra is essentially the study of the rings occurring in
algebraic number theory and
algebraic geometry.
In algebraic number theory, the rings of
algebraic integer
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s are
Dedekind ring
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 ...
s, which constitute therefore an important class of commutative rings. Considerations related to
modular arithmetic have led to the notion of a
valuation ring. The restriction of
algebraic field extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
s to subrings has led to the notions of
integral extensions and
integrally closed domain
In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root ...
s as well as the notion of
ramification of an extension of valuation rings.
The notion of
localization of a ring (in particular the localization with respect to a
prime ideal, the localization consisting in inverting a single element and the
total quotient ring
In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction embeds ...
) is one of the main differences between commutative algebra and the theory of non-commutative rings. It leads to an important class of commutative rings, the
local rings that have only one
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
. The set of the prime ideals of a commutative ring is naturally equipped with a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
. All these notions are widely used in algebraic geometry and are the basic technical tools for the definition of
scheme theory, a generalization of algebraic geometry introduced by
Grothendieck.
Many other notions of commutative algebra are counterparts of geometrical notions occurring in algebraic geometry. This is the case of
Krull dimension
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 th ...
,
primary decomposition
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'' (which are relate ...
,
regular 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 a Noetherian local ring with maximal ide ...
s,
Cohen–Macaulay rings,
Gorenstein ring In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring ''R'' with finite injective dimension as an ''R''-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is ...
s and many other notions.
History
The subject, first known as
ideal theory
In mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only consid ...
, began with
Richard Dedekind's work on
ideals, itself based on the earlier work of
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned ...
and
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
. 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
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
. In turn, Hilbert strongly influenced
Emmy Noether
Amalie Emmy NoetherEmmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
, 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
Emanuel Lasker (; December 24, 1868 – January 11, 1941) was a German chess player, mathematician, and philosopher who was World Chess Champion for 27 years, from 1894 to 1921, the longest reign of any officially recognised World Chess Cham ...
, who introduced
primary ideal In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y'n'' is also an element of ''Q'', for some ''n'' > 0. Fo ...
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
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
, who introduced the fundamental notions of
localization and
completion of a ring, as well as that of
regular local 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 a Noetherian local ring with maximal ide ...
s. He established the concept of the
Krull dimension
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 th ...
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
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian ring. The theorem is sometimes referred to by its German name, ''Krull ...
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
In
mathematics, more specifically in the area of
modern algebra known as
ring theory
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 r ...
, a Noetherian ring, named after
Emmy Noether
Amalie Emmy NoetherEmmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
, is a ring in which every non-empty set of
ideals has a maximal element. Equivalently, a ring is Noetherian if it satisfies the
ascending chain condition on ideals; that is, given any chain:
:
there exists an ''n'' such that:
:
For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. (The result is due to
I. S. Cohen.)
The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of
integers and the
polynomial ring over a
field are both Noetherian rings, and consequently, such theorems as the
Lasker–Noether theorem, the
Krull intersection theorem In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic nu ...
, and the
Hilbert's basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the
descending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...
on ''
prime ideals''. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the
Krull dimension
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 th ...
.
Hilbert's basis theorem
Hilbert's basis theorem has some immediate corollaries:
#By induction we see that