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| {{about|the branch of algebra that studies commutative rings|algebras that are commutative|Commutative algebra (structure)}}
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| [[File:Emmy noether postcard 1915.jpg|thumb|A 1915 postcard from one of the pioneers of commutative algebra, [[Emmy Noether]], to E. Fischer, discussing her work in commutative algebra.]] | | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
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| '''Commutative algebra''' is the branch of [[algebra]] that studies [[commutative ring]]s, their [[ideal (ring theory)|ideals]], and [[module (mathematics)|modules]] over such rings. Both [[algebraic geometry]] and [[algebraic number theory]] build on commutative algebra. Prominent examples of commutative rings include [[polynomial ring]]s, rings of [[algebraic integer]]s, including the ordinary [[integer]]s <math>\mathbb{Z}</math>, and [[p-adic number|''p''-adic integer]]s.<ref>Atiyah and Macdonald, 1969, Chapter 1</ref>
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| Commutative algebra is the main technical tool in the local study of [[scheme (mathematics)|schemes]].
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| The study of rings which are not necessarily commutative is known as [[noncommutative algebra]]; it includes [[ring theory]], [[representation theory]], and the theory of [[Banach algebra]]s.
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| ==Overview== | | '''source''' |
| Commutative algebra is essentially the study of the rings occurring in [[algebraic number theory]] and [[algebraic geometry]]
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| In algebraic number theory, the rings of [[algebraic integer]]s are [[Dedekind ring]]s, which constitute therefore an important class of commutative rings. Considerations related to [[modular arithmetic]] have led to the notion of [[valuation ring]]. The restriction of [[algebraic field extension]]s to subrings has led to the notions of [[integral extension]]s and [[integrally closed domain]]s as well as the notion of [[ramification]] of an extension of valuation rings.
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| 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]]) 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 ring]]s that have only one [[maximal ideal]]. The set of the prime ideals of a commutative ring is naturally equipped with a [[topological space|topology]], the [[Zariski topology]]. 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]].
| | ==Demos== |
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| Many other notions of commutative algebra are counterparts of geometrical notions occurring in algebraic geometry. This is the case of [[Krull dimension]], [[primary decomposition]], [[regular ring]]s, [[Cohen–Macaulay ring]]s, [[Gorenstein ring]]s and many other notions.
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| == History ==
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| 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]].
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| 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.
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| 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 [[Kronecker|Kronecker's]] work, the modern approach to commutative algebra using module theory is usually credited to [[Wolfgang Krull|Krull]] and [[Emmy Noether|Noether]].
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| ==Main tools and results== | | ==Test pages == |
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| ===Noetherian rings===
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| {{Main|Noetherian ring}}
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| In [[mathematics]], more specifically in the area of [[Abstract algebra|modern algebra]] known as [[Ring (mathematics)|ring theory]], a '''Noetherian ring''', named after [[Emmy Noether]], is a ring in which every non-empty set of [[ideal (ring theory)|ideal]]s has a maximal element. Equivalently, a ring is Noetherian if it satisfies the [[ascending chain condition]] on ideals; that is, given any chain:
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| :<math>I_1\subseteq\cdots I_{k-1}\subseteq I_{k}\subseteq I_{k+1}\subseteq\cdots</math>
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| there exists an ''n'' such that:
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| :<math>I_{n}=I_{n+1}=\cdots</math>
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| 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]].)
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| 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 [[integer]]s and the [[polynomial ring]] over a [[Field (mathematics)|field]] are both Noetherian rings, and consequently, such theorems as the [[Lasker–Noether theorem]], the [[Krull intersection theorem]], and the [[Hilbert's basis theorem]] hold for them. Furthermore, if a ring is Noetherian, then it satisfies the [[descending chain condition]] on ''[[prime ideal]]s''. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the [[Krull dimension]].
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| ===Hilbert's basis theorem===
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| {{Main|Hilbert's basis theorem}}
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| <blockquote>'''Theorem.''' If ''R'' is a left (resp. right) [[Noetherian ring]], then the [[polynomial ring]] ''R''[''X''] is also a left (resp. right) Noetherian ring.</blockquote>
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| Hilbert's basis theorem has some immediate corollaries:
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| #By induction we see that <math>R[X_0,\dotsc,X_{n-1}]</math> will also be Noetherian.
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| #Since any [[affine variety]] over <math>R^n</math> (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal <math>\mathfrak a\subset R[X_0, \dotsc, X_{n-1}]</math> and further as the locus of its generators, it follows that every affine variety is the locus of finitely many polynomials — i.e. the intersection of finitely many [[hypersurface]]s.
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| #If <math>A</math> is a finitely-generated <math>R</math>-algebra, then we know that <math>A \simeq R[X_0, \dotsc, X_{n-1}] / \mathfrak a</math>, where <math>\mathfrak a</math> is an ideal. The basis theorem implies that <math>\mathfrak a</math> must be finitely generated, say <math>\mathfrak a = (p_0,\dotsc, p_{N-1})</math>, i.e. <math>A</math> is [[Glossary_of_ring_theory#Finitely_presented_algebra|finitely presented]].
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| ===Primary decomposition===
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| {{Main|Primary decomposition}}
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| An ideal ''Q'' of a ring is said to be ''[[Primary ideal|primary]]'' if ''Q'' is [[proper subset|proper]] and whenever ''xy'' ∈ ''Q'', either ''x'' ∈ ''Q'' or ''y<sup>n</sup>'' ∈ ''Q'' for some positive integer ''n''. In '''Z''', the primary ideals are precisely the ideals of the form (''p<sup>e</sup>'') 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.
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| The ''[[Lasker–Noether theorem]]'', given here, may be seen as a certain generalization of the fundamental theorem of arithmetic:
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| <blockquote>'''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 [[Radical of an ideal|radicals]]; that is:
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| : <math>I=\bigcap_{i=1}^t Q_i</math>
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| with ''Q<sub>i</sub>'' primary for all ''i'' and Rad(''Q<sub>i</sub>'') ≠ Rad(''Q<sub>j</sub>'') for ''i'' ≠ ''j''. Furthermore, if:
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| : <math>I=\bigcap_{i=1}^k P_i</math>
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| is decomposition of ''I'' with Rad(''P<sub>i</sub>'') ≠ Rad(''P<sub>j</sub>'') for ''i'' ≠ ''j'', and both decompositions of ''I'' are ''irredundant'' (meaning that no proper subset of either {''Q''<sub>1</sub>, ..., ''Q<sub>t</sub>''} or {''P''<sub>1</sub>, ..., ''P<sub>k</sub>''} yields an intersection equal to ''I''), ''t'' = ''k'' and (after possibly renumbering the ''Q<sub>i</sub>'') Rad(''Q<sub>i</sub>'') = Rad(''P<sub>i</sub>'') for all ''i''.</blockquote>
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| For any primary decomposition of ''I'', the set of all radicals, that is, the set {Rad(''Q''<sub>1</sub>), ..., Rad(''Q<sub>t</sub>'')} remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the [[associated prime|assassinator]] of the module ''R''/''I''; that is, the set of all [[annihilator (ring theory)|annihilators]] of ''R''/''I'' (viewed as a module over ''R'') that are prime.
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| ===Localization=== | |
| {{main|Localization (algebra)}}
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| The [[localization (algebra)|localization]] is a formal way to introduce the "denominators" to given a ring or a module. That is, it introduces a new ring/module out of an existing one so that it consists of [[algebraic fraction|fractions]]
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| :<math>\frac{m}{s}</math>.
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| where the [[denominator]]s ''s'' range in a given subset ''S'' of ''R''. The basic example is the construction of the ring '''Q''' of rational numbers from the ring '''Z''' of rational integers.
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| ===Completion===
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| {{main|Completion (ring theory)}}
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| A [[completion (ring theory)|completion]] is any of several related [[functor]]s on [[ring (mathematics)|ring]]s and [[module (mathematics)|modules]] that result in complete [[topological ring]]s and modules. Completion is similar to [[localization of a ring|localization]], and together they are among the most basic tools in analysing [[commutative ring]]s. Complete commutative rings have simpler structure than the general ones and [[Hensel's lemma]] applies to them.
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| ===Zariski topology on prime ideals===
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| {{main|Zariski topology}}
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| The [[Zariski topology]] defines a [[topological space|topology]] on the [[spectrum of a ring]] (the set of prime ideals).<ref>{{cite book
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| | last1 = Dummit
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| | first1 = D. S.
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| | last2 = Foote
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| | first2 = R.
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| | title = Abstract Algebra
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| | publisher = Wiley
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| | pages = 71–72
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| | year = 2004
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| | edition = 3
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| | isbn = 9780471433347
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| }}</ref> In this formulation, the Zariski-closed sets are taken to be the sets
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| :<math>V(I) = \{P \in \operatorname{Spec}\,(A) \mid I \subseteq P\}</math> | |
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| 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 (''a<sub>1</sub>'', ..., ''a<sub>n</sub>'') such that (''x<sub>1</sub>'' - ''a<sub>1</sub>'', ..., ''x<sub>n</sub>'' - ''a<sub>n</sub>'') 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.
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| ==Examples== | |
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| The fundamental example in commutative algebra is the ring of integers <math>\mathbb{Z}</math>. The existence of primes and
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| the unique factorization theorem laid the foundations for concepts such as [[Noetherian ring]]s and the [[primary decomposition]].
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| Other important examples are:
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| *[[Polynomial ring]]s <math>R[x_1,...,x_n]</math>
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| *The [[p-adic integer]]s
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| *Rings of [[algebraic integer]]s.
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| ==Connections with algebraic geometry==
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| Commutative algebra (in the form of [[polynomial ring]]s and their quotients, used in the definition of [[algebraic varieties]]) has always been a part of [[algebraic geometry]]. However, in late 1950s, algebraic varieties were subsumed into [[Alexander Grothendieck]]'s concept of a [[scheme (mathematics)|scheme]]. Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent (dual) to the category of commutative unital rings, extending the [[duality (category theory)|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 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 became prominent including [[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 stack]]s, both often called algebraic stacks.
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| == See also ==
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| * [[List of commutative algebra topics]]
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| * [[Glossary of commutative algebra]]
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| * [[Combinatorial commutative algebra]]
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| * [[Gröbner basis]]
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| * [[Homological algebra]]
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| == References ==
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| {{reflist}}
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| * [[Michael Atiyah]] & [[Ian G. Macdonald]], ''[[Introduction to Commutative Algebra]]'', Massachusetts : Addison-Wesley Publishing, 1969.
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| * [[Nicolas Bourbaki|Bourbaki, Nicolas]], ''Commutative algebra. Chapters 1--7''. Translated from the French. Reprint of the 1989 English translation. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998. xxiv+625 pp. ISBN 3-540-64239-0
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| * [[Nicolas Bourbaki|Bourbaki, Nicolas]], ''Éléments de mathématique. Algèbre commutative. Chapitres 8 et 9''. (Elements of mathematics. Commutative algebra. Chapters 8 and 9) Reprint of the 1983 original. Springer, Berlin, 2006. ii+200 pp. ISBN 978-3-540-33942-7
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| * [[David Eisenbud]], ''[[Commutative Algebra With a View Toward Algebraic Geometry]]'', New York : Springer-Verlag, 1999.
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| * Rémi Goblot, "Algèbre commutative, cours et exercices corrigés", 2e édition, Dunod 2001, ISBN 2-10-005779-0
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| * Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1
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| * Matsumura, Hideyuki, ''Commutative algebra''. Second edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. xv+313 pp. ISBN 0-8053-7026-9
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| * Matsumura, Hideyuki, ''Commutative Ring Theory''. Second edition. Translated from the Japanese. Cambridge Studies in Advanced Mathematics, Cambridge, UK : Cambridge University Press, 1989. ISBN 0-521-36764-6
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| * [[Masayoshi Nagata|Nagata, Masayoshi]], ''Local rings''. Interscience Tracts in Pure and Applied Mathematics, No. 13. Interscience Publishers a division of John Wiley and Sons, New York-London 1962 xiii+234 pp.
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| * Miles Reid, ''[[Undergraduate Commutative Algebra]] (London Mathematical Society Student Texts)'', Cambridge, UK : Cambridge University Press, 1996.
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| * [[Jean-Pierre Serre]], ''Local algebra''. Translated from the French by CheeWhye Chin and revised by the author. (Original title: ''Algèbre locale, multiplicités'') Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000. xiv+128 pp. ISBN 3-540-66641-9
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| * Sharp, R. Y., ''Steps in commutative algebra''. Second edition. London Mathematical Society Student Texts, 51. Cambridge University Press, Cambridge, 2000. xii+355 pp. ISBN 0-521-64623-5
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| * [[Oscar Zariski|Zariski, Oscar]]; [[Pierre Samuel|Samuel, Pierre]], ''Commutative algebra''. Vol. 1, 2. With the cooperation of I. S. Cohen. Corrected reprinting of the 1958, 1960 edition. Graduate Texts in Mathematics, No. 28, 29. Springer-Verlag, New York-Heidelberg-Berlin, 1975.
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| * {{Citation | last1=Zeidler | first1=A. Bernhard | title=Abstract Algebra | year=2014 | url=https://wuala.com/Chronicler/Mathematics/algebra.pdf }} A web-book on algebra and commutative algebra. Warning: work in progress! Free downloadable PDF under Open Publication License.
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| [[Category:Commutative algebra| ]]
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