Category: Commutative algebra

Stanley decomposition
In commutative algebra, a Stanley decomposition is a way of writing a ring in terms of polynomial subrings. They were introduced by Richard Stanley.
Mori domain
In algebra, a Mori domain, named after Yoshiro Mori by Querré , is an integral domain satisfying the ascending chain condition on integral divisorial ideals. Noetherian domains and Krull domains both
Tensor-hom adjunction
In mathematics, the tensor-hom adjunction is that the tensor product and hom-functor form an adjoint pair: This is made more precise below. The order of terms in the phrase "tensor-hom adjunction" ref
Associated prime
In abstract algebra, an associated prime of a module M over a ring R is a type of prime ideal of R that arises as an annihilator of a (prime) submodule of M. The set of associated primes is usually de
Auslander–Buchsbaum theorem
In commutative algebra, the Auslander–Buchsbaum theorem states that regular local rings are unique factorization domains. The theorem was first proved by Maurice Auslander and David Buchsbaum. They sh
Dualizing module
In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety. It is used in Grothendieck l
Noetherian module
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. Historically, Hilbert was
Atomic domain
In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain in which every non-zero non-unit can be written in at least one way as a finite product of
Differential graded algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
Koszul–Tate resolution
In mathematics, a Koszul–Tate resolution or Koszul–Tate complex of the quotient ring R/M is a projective resolution of it as an R-module which also has a structure of a dg-algebra over R, where R is a
Primal ideal
In mathematics, an element a of a commutative ring A is called (relatively) prime to an ideal Q if whenever ab is an element of Q then b is also an element of Q. A proper ideal Q of a commutative ring
Irrelevant ideal
In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. More generally, a homogeneous ideal of a graded ring is called an
Hausdorff completion
In algebra, the Hausdorff completion of a group G with filtration is the inverse limit of the discrete group . A basic example is a profinite completion. The image of the canonical map is a Hausdorff
Polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indetermin
Analytically unramified ring
In algebra, an analytically unramified ring is a local ring whose completion is reduced (has no nonzero nilpotent). The following rings are analytically unramified: * pseudo-geometric reduced ring.
Quasi-homogeneous polynomial
In algebra, a multivariate polynomial is quasi-homogeneous or weighted homogeneous, if there exist r integers , called weights of the variables, such that the sum is the same for all nonzero terms of
Stanley–Reisner ring
In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finit
Differential calculus over commutative algebras
In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formul
Principal ideal ring
In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by
Cluster algebra
Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky . A cluster algebra of rank n is an integral domain A, together with some subsets of size n called clusters whose u
Ideal norm
In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of
Hilbert's syzygy theorem
In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important
Rees decomposition
In commutative algebra, a Rees decomposition is a way of writing a ring in terms of polynomial subrings. They were introduced by David Rees.
Weierstrass ring
In mathematics, a Weierstrass ring, named by Nagata after Karl Weierstrass, is a commutative local ring that is Henselian, pseudo-geometric, and such that any quotient ring by a prime ideal is a finit
Nilradical of a ring
In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements: In the non-commutative ring case the same definition does not always work. This has resulted in seve
Laurent polynomial
In mathematics, a Laurent polynomial (namedafter Pierre Alphonse Laurent) in one variable over a field is a linear combination of positive and negative powers of the variable with coefficients in . La
Matlis duality
In algebra, Matlis duality is a duality between Artinian and Noetherian modules over a complete Noetherian local ring. In the special case when the local ring has a field mapping to the residue field
Artinian ideal
In abstract algebra, an Artinian ideal, named after Emil Artin, is encountered in ring theory, in particular, with polynomial rings. Given a polynomial ring R = k[X1, ... Xn] where k is some field, an
Henselian ring
In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings
Linear relation
In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution. More precisely, if are elements of
Tensor product of algebras
In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of s
Regular ring
No description available.
Finite algebra
In abstract algebra, an -algebra is finite if it is finitely generated as an -module. An -algebra can be thought as a homomorphism of rings , in this case is called a finite morphism if is a finite -a
Homological conjectures in commutative algebra
In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectur
Grothendieck local duality
In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves.
Total ring of fractions
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
Completion of a ring
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 the
Faltings' annihilator theorem
In abstract algebra (specifically commutative ring theory), Faltings' annihilator theorem states: given a finitely generated module M over a Noetherian commutative ring A and ideals I, J, the followin
Fitting ideal
In commutative algebra, the Fitting ideals of a finitely generated module over a commutative ring describe the obstructions to generating the module by a given number of elements. They were introduced
Divided domain
In algebra, a divided domain is an integral domain R in which every prime ideal satisfies . A locally divided domain is an integral domain that is a divided domain at every maximal ideal. A Prüfer dom
Catenary ring
In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains p=p0 ⊂p1 ... ⊂pn= q of prime ideals are contained in maximal strictly increasi
Integer-valued polynomial
In mathematics, an integer-valued polynomial (also known as a numerical polynomial) is a polynomial whose value is an integer for every integer n. Every polynomial with integer coefficients is integer
Glossary of commutative algebra
This is a glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary of ring theory and glossary of
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of th
Rees algebra
In commutative algebra, the Rees algebra of an ideal I in a commutative ring R is defined to be The extended Rees algebra of I (which some authors refer to as the Rees algebra of I) is defined as This
André–Quillen cohomology
In commutative algebra, André–Quillen cohomology is a theory of cohomology for commutative rings which is closely related to the cotangent complex. The first three cohomology groups were introduced by
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 ev
Cohen ring
In algebra, a Cohen ring is a field or a complete discrete valuation ring of mixed characteristic whose maximal ideal is generated by p. Cohen rings are used in the Cohen structure theorem for complet
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3.
Serre's multiplicity conjectures
In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since Andr
Hilbert–Poincaré series
In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the n
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
Hilbert–Samuel function
In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module over a commutative Noetherian local ring and a primary ideal of
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 finit
Unibranch local ring
In algebraic geometry, a local ring A is said to be unibranch if the reduced ring Ared (obtained by quotienting A by its nilradical) is an integral domain, and the integral closure B of Ared is also a
Weak dimension
In abstract algebra, the weak dimension of a nonzero right module M over a ring R is the largest number n such that the Tor group is nonzero for some left R-module N (or infinity if no largest such n
Cohen–Macaulay 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
Regular sequence
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric not
Ascending 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 commutativ
Eakin–Nagata theorem
In abstract algebra, the Eakin–Nagata theorem states: given commutative rings such that is finitely generated as a module over , if is a Noetherian ring, then is a Noetherian ring. (Note the converse
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 commutativ
Test ideal
A test ideal is a positive characteristic analog of a multiplier ideal in, say, the field of complex numbers. Test ideals are used in the study of singularities in algebraic geometry in positive chara
Weierstrass preparation theorem
In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It states that such a function is, up to multiplicat
Monomial conjecture
In commutative algebra, a field of mathematics, the monomial conjecture of Melvin Hochster says the following: Let A be a Noetherian local ring of Krull dimension d and let x1, ..., xd be a system of
Zariski ring
In commutative algebra, a Zariski ring is a commutative Noetherian topological ring A whose topology is defined by an ideal contained in the Jacobson radical, the intersection of all maximal ideals. T
Cohen structure theorem
In mathematics, the Cohen structure theorem, introduced by Cohen, describes the structure of complete Noetherian local rings. Some consequences of Cohen's structure theorem include three conjectures o
Local parameter
In the geometry of complex algebraic curves, a local parameter for a curve C at a smooth point P is just a meromorphic function on C that has a simple zero at P. This concept can be generalized to cur
J-2 ring
In commutative algebra, a J-0 ring is a ring such that the set of regular points, that is, points of the spectrum at which the localization is a regular local ring, contains a non-empty open subset, a
Linear equation over a ring
In algebra, linear equations and systems of linear equations over a field are widely studied. "Over a field" means that the coefficients of the equations and the solutions that one is looking for belo
Dimension theory (algebra)
In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an appa
Discrete valuation
In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function: satisfying the conditions: for all . Note that often the trivial valuation which takes on only the value
Change of rings
In algebra, given a ring homomorphism , there are three ways to change the coefficient ring of a module; namely, for a left R-module M and a left S-module N, * , the induced module. * , the coinduce
Conductor (ring theory)
In ring theory, a branch of mathematics, the conductor is a measurement of how far apart a commutative ring and an extension ring are. Most often, the larger ring is a domain integrally closed in its
Constructible topology
In commutative algebra, the constructible topology on the spectrum of a commutative ring is a topology where each closed set is the image of in for some algebra B over A. An important feature of this
Essential extension
In mathematics, specifically module theory, given a ring R and an R-module M with a submodule N, the module M is said to be an essential extension of N (or N is said to be an essential submodule or la
Analytically normal ring
In algebra, an analytically normal ring is a local ring whose completion is a normal ring, in other words a domain that is integrally closed in its quotient field. proved that if a local ring of an al
Kähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as st
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 yn is also an element of Q, for some n > 0.
Buchberger's algorithm
In the theory of multivariate polynomials, Buchberger's algorithm is a method for transforming a given set of polynomials into a Gröbner basis, which is another set of polynomials that have the same c
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 somet
Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a which allows a suitable generalization of the Eucli
Local cohomology
In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES writt
Ideal reduction
The reduction theory goes back to the influential 1954 paper by Northcott and Rees, the paper that introduced the basic notions. In algebraic geometry, the theory is among the essential tools to extra
Multiplicatively closed set
In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset S of a ring R such that the following two conditions hold: * , * for all . In other words, S is closed under ta
Jacobson ring
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
Arf ring
In mathematics, an Arf ring was defined by to be a 1-dimensional commutative semi-local Macaulay ring satisfying some extra conditions studied by Cahit Arf.
Artin–Rees lemma
In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by th
Formally smooth map
In algebraic geometry and commutative algebra, a ring homomorphism is called formally smooth (from French: Formellement lisse) if it satisfies the following infinitesimal lifting property: Suppose B i
Generic flatness
In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free. They are due t
Excellent ring
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally
Field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between t
Invariant basis number
In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over R have a well-defined rank. In the
Integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the
Valuation ring
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x−1 belongs to D. Given a field F, if D is a subring of F
Prüfer domain
In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedek
Gröbner fan
In computer algebra, the Gröbner fan of an ideal in the ring of polynomials is a concept in the theory of Gröbner bases. It is defined to be a fan consisting of cones that correspond to different mono
Auslander–Buchsbaum formula
In commutative algebra, the Auslander–Buchsbaum formula, introduced by Auslander and Buchsbaum , states that if R is a commutative Noetherian local ring and M is a non-zero finitely generated R-module
Bass–Quillen conjecture
In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring . The conjecture is named for Hyman Bass and Daniel Quillen, who formul
I-adic topology
In commutative algebra, the mathematical study of commutative rings, adic topologies are a family of topologies on elements of a module, generalizing the p-adic topologies on the integers.
Depth (ring theory)
In commutative and homological algebra, depth is an important invariant of rings and modules. Although depth can be defined more generally, the most common case considered is the case of modules over
Locally nilpotent
In the mathematical field of commutative algebra, an ideal I in a commutative ring A is locally nilpotent at a prime ideal p if Ip, the localization of I at p, is a nilpotent ideal in Ap. In non-commu
Simplicial commutative ring
In algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If A is a si
Abhyankar's inequality
Abhyankar's inequality is an inequality involving extensions of valued fields in algebra, introduced by Abhyankar. If K/k is an extension of valued fields, then Abhyankar's inequality states that the
Serre–Swan theorem
In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules a
Dual number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form a + bε, where a and b are real numbers, and ε is a symbol taken to
Dedekind domain
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
Puiseux series
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series is a Puiseux series in the indetermin
Bézout domain
In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a
FGLM algorithm
FGLM is one of the main algorithms in computer algebra, named after its designers, Faugère, Gianni, Lazard and Mora. They introduced their algorithm in 1993. The input of the algorithm is a Gröbner ba
Parafactorial local ring
In algebraic geometry, a Noetherian local ring R is called parafactorial if it has depth at least 2 and the Picard group Pic(Spec(R) − m) of its spectrum with the closed point m removed is trivial. Mo
Zero-divisor graph
In mathematics, and more specifically in combinatorial commutative algebra, a zero-divisor graph is an undirected graph representing the zero divisors of a commutative ring. It has elements of the rin
Seminormal ring
In algebra, a seminormal ring is a commutative reduced ring in which, whenever x, y satisfy , there is s with and . This definition was given by as a simplification of the original definition of . A b
Going up and going down
In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions. The phrase going up refers to the
Mori–Nagata theorem
In algebra, the Mori–Nagata theorem introduced by Yoshiro Mori and Nagata, states the following: let A be a noetherian reduced commutative ring with the total ring of fractions K. Then the integral cl
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 o
Ring of mixed characteristic
In commutative algebra, a ring of mixed characteristic is a commutative ring having characteristic zero and having an ideal such that has positive characteristic.
Combinatorial commutative algebra
Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra
Noether normalization lemma
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
Hahn series
In mathematics, Hahn series (sometimes also known as Hahn–Mal'cev–Neumann series) are a type of formal infinite series. They are a generalization of Puiseux series (themselves a generalization of form
Gröbner basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal i
System of parameters
In mathematics, a system of parameters for a local Noetherian ring of Krull dimension d with maximal ideal m is a set of elements x1, ..., xd that satisfies any of the following equivalent conditions:
Hironaka decomposition
In mathematics, a Hironaka decomposition is a representation of an algebra over a field as a finitely generated free module over a polynomial subalgebra or a regular local ring. Such decompositions ar
Invariant polynomial
In mathematics, an invariant polynomial is a polynomial that is invariant under a group acting on a vector space . Therefore, is a -invariant polynomial if for all and . Cases of particular importance
Nakayama's lemma
In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring
Congruence ideal
In algebra, the congruence ideal of a surjective ring homomorphism f : B → C of commutative rings is the image under f of the annihilator of the kernel of f. It is called a congruence ideal because wh
Principal ideal
In mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of The term also has another, similar
Commutative ring
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
Difference of two squares
In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity i
Tight closure
In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by Melvin Hochster and Craig Huneke . Let be a commuta
Almost ring
In mathematics, almost modules and almost rings are certain objects interpolating between rings and their fields of fractions. They were introduced by Gerd Faltings in his study of p-adic Hodge theory
Krull ring
In commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They are a higher-dimens
Bass number
In mathematics, the ith Bass number of a module M over a local ring R with residue field k is the k-dimension of . More generally the Bass number of a module M over a ring R at a prime ideal p is the
Minimal prime ideal
In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principa
Étale algebra
In commutative algebra, an étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite separable field extensions. These may also be called separable
GCD domain
In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal
Buchsbaum ring
In mathematics, Buchsbaum rings are Noetherian local rings such that every system of parameters is a weak sequence.A sequence of the maximal ideal is called a weak sequence if for all . They were intr
Quasi-unmixed ring
In algebra, specifically in the theory of commutative rings, a quasi-unmixed ring (also called a formally equidimensional ring in EGA) is a Noetherian ring such that for each prime ideal p, the comple
Schubert variety
In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain
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 theor
Complete intersection ring
In commutative algebra, a complete intersection ring is a commutative ring similar to the coordinate rings of varieties that are complete intersections. Informally, they can be thought of roughly as t
Novikov ring
In mathematics, given an additive subgroup , the Novikov ring of is the subring of consisting of formal sums such that and . The notion was introduced by Sergei Novikov in the papers that initiated th
Integral element
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 aj in A such that That is to say, b is a root of a monic polynomial o
Local criterion for flatness
In algebra, the local criterion for flatness gives conditions one can check to show flatness of a module.
Irreducible ring
In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways. * A (meet-)irreducible ring is one in which the intersection of two nonzero ideals i
Finitely generated algebra
In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra A over a field K where there exists a finite set of elements a1,...,an of A su
Zariski's finiteness theorem
In algebra, Zariski's finiteness theorem gives a positive answer to Hilbert's 14th problem for the polynomial ring in two variables, as a special case. Precisely, it states: Given a normal domain A, f
Artin approximation theorem
In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin in deformation theory which implies that formal power series with coefficients in a field k are well-approximat
Principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonze
Quillen–Suslin theorem
The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polyno
Multiplier ideal
In commutative algebra, the multiplier ideal associated to a sheaf of ideals over a complex variety and a real number c consists (locally) of the functions h such that is locally integrable, where the
Wu's method of characteristic set
Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept
Introduction to Commutative Algebra
Introduction to Commutative Algebra is a well-known commutative algebra textbook written by Michael Atiyah and Ian G. Macdonald. It deals with elementary concepts of commutative algebra including loca
List of commutative algebra topics
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutativ
In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined below). Almost all Noetherian rings that oc
Hilbert–Burch theorem
In mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient has projective dimension 2. Hilbert pro
Analytically irreducible ring
In algebra, an analytically irreducible ring is a local ring whose completion has no zero divisors. Geometrically this corresponds to a variety with only one analytic branch at a point. proved that if
Hilbert's basis theorem
In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.
Artin–Tate lemma
In algebra, the Artin–Tate lemma, named after Emil Artin and John Tate, states: Let A be a commutative Noetherian ring and commutative algebras over A. If C is of finite type over A and if C is finite
Divided power structure
In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form meaningful even when it is not possible to actually divide by .
Connected ring
In mathematics, especially in the field of commutative algebra, a connected ring is a commutative ring A that satisfies one of the following equivalent conditions: * A possesses no non-trivial (that
Geometrically regular ring
In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. Geometrically regular schemes are define
Discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain R which satisfies any one of the
Complete intersection
In mathematics, an algebraic variety V in projective space is a complete intersection if the ideal of V is generated by exactly codim V elements. That is, if V has dimension m and lies in projective s
Nagata ring
In commutative algebra, an N-1 ring is an integral domain whose integral closure in its quotient field is a finitely generated -module. It is called a Japanese ring (or an N-2 ring) if for every finit
Integral closure of an ideal
In algebra, the integral closure of an ideal I of a commutative ring R, denoted by , is the set of all elements r in R that are integral over I: there exist such that It is similar to the integral clo
Serre's inequality on height
In algebra, specifically in the theory of commutative rings, Serre's inequality on height states: given a (Noetherian) regular ring A and a pair of prime ideals in it, for each prime ideal that is a m
Rabinowitsch trick
In mathematics, the Rabinowitsch trick, introduced by George Yuri Rainich and published under his original name ,is a short way of proving the general case of the Hilbert Nullstellensatz from an easie
Acceptable ring
In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein ring
Deviation of a local ring
In commutative algebra, the deviations of a local ring R are certain invariants εi(R) that measure how far the ring is from being regular.
Top (algebra)
In the context of a module M over a ring R, the top of M is the largest semisimple quotient module of M if it exists. For finite-dimensional k-algebras (k a field) R, if rad(M) denotes the intersectio
Spectrum of a ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring R is the set of all prime ideals of R, and is usually denoted by ; in algebraic geometry it is simultaneously a topologica
Hensel's lemma
In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a pri
In algebra, a j-multiplicity is a generalization of a Hilbert–Samuel multiplicity. For m-primary ideals, the two notions coincide.
Hodge algebra
In mathematics, a Hodge algebra or algebra with straightening law is a commutative algebra that is a free module over some ring R, together with a given basis similar to the basis of standard monomial
Hilbert series and Hilbert polynomial
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which me