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79 relations: Addison-Wesley, Algebraic extension, Algebraic geometry, Algebraic independence, Algebraic integer, Algebraic number theory, Analytically unramified ring, Annihilator (ring theory), Éléments de mathématique, Cambridge University Press, Cayley–Hamilton theorem, Commutative algebra, Commutative ring, Constructible set (topology), Cyclotomic field, David Eisenbud, Determinant, Direct limit, Endomorphism, Field (mathematics), Field extension, Finite group, Finite morphism, Finitely generated module, Fixed-point subring, Gaussian integer, Glossary of algebraic geometry, Going up and going down, Graded ring, Henselian ring, Homogeneous coordinate ring, Ian G. Macdonald, Ideal (ring theory), Idealizer, Idempotent (ring theory), Integral closure of an ideal, Integrally closed domain, James Milne (mathematician), Krull dimension, Krull ring, Krull–Akizuki theorem, Localization of a ring, Maximal ideal, Michael Atiyah, Miles Reid, Minimal prime ideal, Monic polynomial, Mori–Nagata theorem, Multiplicatively closed set, Nagata ring, ..., Nakayama's lemma, Nicolas Bourbaki, Nilpotent, Noether normalization lemma, Noetherian ring, Normal extension, Normal scheme, Number theory, Open and closed maps, Prime avoidance lemma, Projective variety, Puiseux series, Purely inseparable extension, Quadratic integer, Quotient space (topology), Radical of an ideal, Rational number, Ring (mathematics), Ring homomorphism, Ring of integers, Root of unity, Scheme (mathematics), Subring, Support of a module, Torsor (algebraic geometry), Total ring of fractions, Unibranch local ring, University of Chicago Press, Valuation ring. Expand index (29 more) »
Addison-Wesley
Addison-Wesley is a publisher of textbooks and computer literature.
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Algebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental.
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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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Algebraic independence
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. In particular, a one element set is algebraically independent over K if and only if α is transcendental over K. In general, all the elements of an algebraically independent set S over K are by necessity transcendental over K, and over all of the field extensions over K generated by the remaining elements of S.
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Algebraic integer
In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers).
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Algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.
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Analytically unramified ring
In algebra, an analytically unramified ring is a local ring whose completion is reduced (has no nonzero nilpotent).
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Annihilator (ring theory)
In mathematics, specifically module theory, the annihilator of a set is a concept generalizing torsion and orthogonality.
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Éléments de mathématique
Éléments de mathématique is a treatise on mathematics by the collective Nicolas Bourbaki, composed of twelve books (each divided into one or more chapters).
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Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation.
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Commutative algebra
Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.
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Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
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Constructible set (topology)
In topology, a constructible set in a topological space is a finite union of locally closed sets.
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Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to, the field of rational numbers.
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David Eisenbud
David Eisenbud (born 8 April 1947 in New York City) is an American mathematician.
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Determinant
In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.
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Direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way.
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Endomorphism
In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Field extension
In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F and the operations of F are those of E restricted to F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
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Finite group
In abstract algebra, a finite group is a mathematical group with a finite number of elements.
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Finite morphism
In algebraic geometry, a morphism f: X → Y of schemes is a finite morphism if Y has an open cover by affine schemes such that for each i, is an open affine subscheme Spec Ai, and the restriction of f to Ui, which induces a ring homomorphism makes Ai a finitely generated module over Bi.
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Finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set.
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Fixed-point subring
In algebra, the fixed-point subring R^f of an automorphism f of a ring R is the subring of the fixed points of f: More generally, if G is a group acting on R, then the subring of R: is called the fixed subring or, more traditionally, the ring of invariants.
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Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers.
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Glossary of algebraic geometry
This is a glossary of algebraic geometry.
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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.
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Graded ring
In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_.
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Henselian ring
In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds.
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Homogeneous coordinate ring
In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient ring where I is the homogeneous ideal defining V, K is the algebraically closed field over which V is defined, and is the polynomial ring in N + 1 variables Xi.
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Ian G. Macdonald
Ian Grant Macdonald (born 11 October 1928 in London, England) is a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebra, algebraic combinatorics, and combinatorics.
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Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
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Idealizer
In abstract algebra, the idealizer of a subsemigroup T of a semigroup S is the largest subsemigroup of S in which T is an ideal.
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Idempotent (ring theory)
In abstract algebra, more specifically in ring theory, an idempotent element, or simply an idempotent, of a ring is an element a such that.
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Integral closure of an ideal
In algebra, the integral closure of an ideal I of a commutative ring R, denoted by \overline, is the set of all elements r in R that are integral over I: there exist a_i \in I^i such that It is similar to the integral closure of a subring.
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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.
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James Milne (mathematician)
James S. Milne (born October 10, 1942 in Invercargill, New Zealand) is a New Zealand mathematician working in arithmetic geometry.
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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.
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Krull ring
In commutative algebra, a Krull ring or Krull domain is a commutative ring with a well behaved theory of prime factorization.
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Krull–Akizuki theorem
In algebra, the Krull–Akizuki theorem states the following: let A be a one-dimensional reduced noetherian ring, K its total ring of fractions.
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Localization of a ring
In commutative algebra, localization is a systematic method of adding multiplicative inverses to a ring.
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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.
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Michael Atiyah
Sir Michael Francis Atiyah (born 22 April 1929) is an English mathematician specialising in geometry.
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Miles Reid
Miles Anthony Reid FRS (born 30 January 1948) is a mathematician who works in algebraic geometry.
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Minimal prime ideal
In mathematics, especially in the area of algebra known as commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules.
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Monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.
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Mori–Nagata theorem
In algebra, the Mori–Nagata theorem introduced by and, states the following: let A be a noetherian reduced commutative ring with the total ring of fractions K. Then the integral closure of A in K is a direct product of r Krull domains, where r is the number of minimal prime ideals of A. The theorem is a partial generalization of the Krull–Akizuki theorem, which concerns a one-dimensional noetherian domain.
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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.
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Nagata ring
In commutative algebra, an integral domain A is called an N−1 ring if its integral closure in its quotient field is a finitely generated A module.
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Nakayama's lemma
In mathematics, more specifically modern algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules.
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Nicolas Bourbaki
Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.
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Nilpotent
In mathematics, an element, x, of a ring, R, is called nilpotent if there exists some positive integer, n, such that xn.
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Noether normalization lemma
In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926.
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Noetherian ring
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 chain of left (or right) ideals: there exists an n such that: Noetherian rings are named after Emmy Noether.
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Normal extension
In abstract algebra, an algebraic field extension L/K is said to be normal if every polynomial that is irreducible over K either has no root in L or splits into linear factors in L. Bourbaki calls such an extension a quasi-Galois extension.
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Normal scheme
In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain.
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Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
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Open and closed maps
In topology, an open map is a function between two topological spaces which maps open sets to open sets.
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Prime avoidance lemma
In algebra, the prime avoidance lemma says that if an ideal I in a commutative ring R is contained in a union of finitely many prime ideals Pi's, then it is contained in Pi for some i. There are many variations of the lemma (cf. Hochster); for example, if the ring R contains an infinite field or a finite field of sufficiently large cardinality, then the statement follows from a fact in linear algebra that a vector space over an infinite field or a finite field of large cardinality is not a finite union of its proper vector subspaces.
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Projective variety
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective ''n''-space Pn over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety.
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Puiseux series
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate T. They were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850.
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Purely inseparable extension
In algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of an equation of the form xq.
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Quadratic integer
In number theory, quadratic integers are a generalization of the integers to quadratic fields.
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Quotient space (topology)
In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.
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Radical of an ideal
In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal (or semiprime ideal) is an ideal that is its own radical (this can be phrased as being a fixed point of an operation on ideals called 'radicalization').
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
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Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.
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Ring of integers
In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in.
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Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.
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Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.
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Subring
In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R).
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Support of a module
In commutative algebra, the support of a module M over a commutative ring A is the set of all prime ideals \mathfrak of A such that M_\mathfrak \ne 0.
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Torsor (algebraic geometry)
In algebraic geometry, given a smooth algebraic group G, a G-torsor or a principal G-bundle P over a scheme X is a scheme (or even algebraic space) with an action of G that is locally trivial in the given Grothendieck topology in the sense that the base change Y \times_X P along "some" covering map Y \to X is the trivial torsor Y \times G \to Y (G acts only on the second factor).
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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 have zero divisors.
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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 local ring.
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University of Chicago Press
The University of Chicago Press is the largest and one of the oldest university presses in the United States.
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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 such that either x or x −1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring.
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Complete integral closure, Integral (ring theory), Integral closure, Integral dependence, Integral extension, Integral extension of a ring, Integral ring extension, Integrality, Integrally closed ring.
References
[1] https://en.wikipedia.org/wiki/Integral_element
