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**Commutative algebra** 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] }

- Overview
- History
- Main tools and results
- Noetherian rings
- Hilbert's basis theorem
- Primary decomposition
- Localization
- Completion
- Zariski topology on prime ideals
- Examples
- Connections with algebraic geometry
- See also
- Notes
- References

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, representation theory, and the theory of Banach algebras.

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 integers are Dedekind rings, 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 extensions to subrings has led to the notions of integral extensions and integrally closed domains 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) 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. The set of the prime ideals of a commutative ring is naturally equipped with a 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.

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 rings, Cohen–Macaulay rings, Gorenstein rings and many other notions.

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.

In mathematics, more specifically in the area of modern algebra known as ring theory, a **Noetherian ring**, named after Emmy Noether, 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, and the Hilbert's basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on * prime ideals *. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.

Theorem.IfRis a left (resp. right) Noetherian ring, then the polynomial ringR[X] is also a left (resp. right) Noetherian ring.

Hilbert's basis theorem has some immediate corollaries:

- By induction we see that will also be Noetherian.
- Since any affine variety over (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal 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 hypersurfaces.
- If is a finitely-generated -algebra, then we know that , where is an ideal. The basis theorem implies that must be finitely generated, say , i.e. is finitely presented.

An ideal *Q* of a ring is said to be * primary * if *Q* is proper and whenever *xy* ∈ *Q*, either *x* ∈ *Q* or *y ^{n}* ∈

The * Lasker–Noether theorem *, given here, may be seen as a certain generalization of the fundamental theorem of arithmetic:

Lasker-Noether Theorem.LetRbe a commutative Noetherian ring and letIbe an ideal ofR. ThenImay be written as the intersection of finitely many primary ideals with distinct radicals; that is:

with

Qprimary for all_{i}iand Rad(Q) ≠ Rad(_{i}Q) for_{j}i≠j. Furthermore, if:

is decomposition of

Iwith Rad(P) ≠ Rad(_{i}P) for_{j}i≠j, and both decompositions ofIareirredundant(meaning that no proper subset of either {Q_{1}, ...,Q} or {_{t}P_{1}, ...,P} yields an intersection equal to_{k}I),t=kand (after possibly renumbering theQ) Rad(_{i}Q) = Rad(_{i}P) for all_{i}i.

For any primary decomposition of *I*, the set of all radicals, that is, the set {Rad(*Q*_{1}), ..., Rad(*Q _{t}*)} 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

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.

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.

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 (*a _{1}*, ...,

The fundamental example in commutative algebra is the ring of integers . The existence of primes and the unique factorization theorem laid the foundations for concepts such as Noetherian rings and the primary decomposition.

Other important examples are:

- Polynomial rings
- The p-adic integers
- Rings of algebraic integers.

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.

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

**Algebraic geometry** is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

In mathematics, **rings** are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, 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, more specifically in the area of abstract algebra known as ring theory, a **Noetherian ring** is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any increasing sequence of left ideals:

In ring theory, a branch of abstract algebra, 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 noncommutative rings where multiplication is not required to be commutative.

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.

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 number fields examined at a particular place, or prime. **Local algebra** is the branch of commutative algebra that studies commutative local rings and their modules.

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, particularly it is not Hausdorff. This topology introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring.

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, a **scheme** is a mathematical 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 mathematics, a **Cohen–Macaulay ring** is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.

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 ideal m, and suppose *a*_{1}, ..., *a*_{n} is a minimal set of generators of m. Then by Krull's principal ideal theorem *n* ≥ dim *A*, and *A* is defined to be regular if *n* = dim *A*.

In mathematics, **ideal theory** is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra. The name grew out of the central considerations, such as the Lasker–Noether theorem in algebraic geometry, and the ideal class group in algebraic number theory, of the commutative algebra of the first quarter of the twentieth century. It was used in the influential van der Waerden text on abstract algebra from around 1930.

In mathematics, the **Noether normalization lemma** is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field *k*, and any finitely generated commutative *k*-algebra*A*, there exists a non-negative integer *d* and algebraically independent elements *y*_{1}, *y*_{2}, ..., *y*_{d} in *A* such that *A* is a finitely generated module over the polynomial ring *S* = *k* [*y*_{1}, *y*_{2}, ..., *y*_{d}].

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 case *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*, a subring of *B*, if there are *n* ≥ 1 and *a*_{j} in *A* such that

This is a **glossary of commutative algebra**.

In algebraic geometry, the **main theorem of elimination theory** states that every projective scheme is proper. A version of this theorem predates the existence of scheme theory. It can be stated, proved, and applied in the following more classical setting. Let *k* be a field, denote by the *n*-dimensional projective space over *k*. The main theorem of elimination theory is the statement that for any *n* and any algebraic variety V defined over *k*, the projection map sends Zariski-closed subsets to Zariski-closed subsets.

- Michael Atiyah & Ian G. Macdonald,
*Introduction to Commutative Algebra*, Massachusetts : Addison-Wesley Publishing, 1969. - 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 - 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 - Eisenbud, David (1995).
*Commutative algebra with a view toward algebraic geometry*. Graduate Texts in Mathematics.**150**. New York: Springer-Verlag. xvi+785. ISBN 0-387-94268-8. MR 1322960. - Rémi Goblot, "Algèbre commutative, cours et exercices corrigés", 2e édition, Dunod 2001, ISBN 2-10-005779-0
- Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1
- 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 - 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 - 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. - Miles Reid,
*Undergraduate Commutative Algebra (London Mathematical Society Student Texts)*, Cambridge, UK : Cambridge University Press, 1996. - 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 - 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 - Zariski, Oscar; 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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