Faster access than browser!
83 relations: A K Peters, Abstract algebra, Affine variety, Algebraic geometry, Algebraic variety, Analytic function, Artinian ring, Cambridge University Press, Cancellation property, Characteristic (algebra), Chinese remainder theorem, Closure (mathematics), Commutative ring, Complex number, Composite number, Connectedness, Continuous function, Dedekind–Hasse norm, Direct limit, Divisibility (ring theory), Domain (ring theory), Elliptic curve, Fiber product of schemes, Field (mathematics), Field of fractions, Finite field, Frobenius endomorphism, Fundamental theorem of arithmetic, GCD domain, Glossary of algebraic geometry, Graduate Studies in Mathematics, Holomorphic function, Ideal (ring theory), Idempotent (ring theory), Injective function, Integer, Integral, Integral domain, Irreducible element, Irreducible polynomial, Irreducible ring, Isomorphism, John Wiley & Sons, Localization of a ring, Manifold, Mathematics, Matrix (mathematics), Matrix ring, Minimal prime ideal, Monoid, ..., Nilradical of a ring, Open set, P-adic number, Polynomial, Primary decomposition, Prime element, Prime ideal, Prime number, Principal ideal, Product ring, Quadratic integer, Quotient ring, Rational number, Real number, Reduced ring, Regular local ring, Ring of integers, Serge Lang, Spectrum of a ring, Springer Science+Business Media, Subclass (set theory), Subring, Tensor product of algebras, Topological space, Unique factorization domain, Unit (ring theory), Unit interval, Wedderburn's little theorem, Zero divisor, Zero element, Zero ring, Zero-product property, 1. Expand index (33 more) »
A K Peters
A K Peters, Ltd. was a publisher of scientific and technical books, specializing in mathematics and in computer graphics, robotics, and other fields of computer science.
New!!: Integral domain and A K Peters · See more »
Abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
New!!: Integral domain and Abstract algebra · See more »
Affine variety
In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine ''n''-space k^n of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal.
New!!: Integral domain and Affine variety · See more »
Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
New!!: Integral domain and Algebraic geometry · See more »
Algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry.
New!!: Integral domain and Algebraic variety · See more »
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series.
New!!: Integral domain and Analytic function · See more »
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals.
New!!: Integral domain and Artinian ring · See more »
Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
New!!: Integral domain and Cambridge University Press · See more »
Cancellation property
In mathematics, the notion of cancellative is a generalization of the notion of invertible.
New!!: Integral domain and Cancellation property · See more »
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
New!!: Integral domain and Characteristic (algebra) · See more »
Chinese remainder theorem
The Chinese remainder theorem is a theorem of number theory, which states that if one knows the remainders of the Euclidean division of an integer by several integers, then one can determine uniquely the remainder of the division of by the product of these integers, under the condition that the divisors are pairwise coprime.
New!!: Integral domain and Chinese remainder theorem · See more »
Closure (mathematics)
A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.
New!!: Integral domain and Closure (mathematics) · See more »
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
New!!: Integral domain and Commutative ring · See more »
Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
New!!: Integral domain and Complex number · See more »
Composite number
A composite number is a positive integer that can be formed by multiplying together two smaller positive integers.
New!!: Integral domain and Composite number · See more »
Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece".
New!!: Integral domain and Connectedness · See more »
Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
New!!: Integral domain and Continuous function · See more »
Dedekind–Hasse norm
In mathematics, in particular the study of abstract algebra, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains.
New!!: Integral domain and Dedekind–Hasse norm · See more »
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.
New!!: Integral domain and Direct limit · See more »
Divisibility (ring theory)
In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers.
New!!: Integral domain and Divisibility (ring theory) · See more »
Domain (ring theory)
In mathematics, and more specifically in algebra, a domain is a nonzero ring in which implies or.
New!!: Integral domain and Domain (ring theory) · See more »
Elliptic curve
In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.
New!!: Integral domain and Elliptic curve · See more »
Fiber product of schemes
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction.
New!!: Integral domain and Fiber product of schemes · See more »
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.
New!!: Integral domain and Field (mathematics) · See more »
Field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded.
New!!: Integral domain and Field of fractions · See more »
Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.
New!!: Integral domain and Finite field · See more »
Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic, an important class which includes finite fields.
New!!: Integral domain and Frobenius endomorphism · See more »
Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.
New!!: Integral domain and Fundamental theorem of arithmetic · See more »
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).
New!!: Integral domain and GCD domain · See more »
Glossary of algebraic geometry
This is a glossary of algebraic geometry.
New!!: Integral domain and Glossary of algebraic geometry · See more »
Graduate Studies in Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS).
New!!: Integral domain and Graduate Studies in Mathematics · See more »
Holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.
New!!: Integral domain and Holomorphic function · See more »
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
New!!: Integral domain and Ideal (ring theory) · See more »
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.
New!!: Integral domain and Idempotent (ring theory) · See more »
Injective function
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.
New!!: Integral domain and Injective function · See more »
Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
New!!: Integral domain and Integer · See more »
Integral
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.
New!!: Integral domain and Integral · See more »
Integral domain
In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.
New!!: Integral domain and Integral domain · See more »
Irreducible element
In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.
New!!: Integral domain and Irreducible element · See more »
Irreducible polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials.
New!!: Integral domain and Irreducible polynomial · See more »
Irreducible ring
In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways.
New!!: Integral domain and Irreducible ring · See more »
Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
New!!: Integral domain and Isomorphism · See more »
John Wiley & Sons
John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.
New!!: Integral domain and John Wiley & Sons · See more »
Localization of a ring
In commutative algebra, localization is a systematic method of adding multiplicative inverses to a ring.
New!!: Integral domain and Localization of a ring · See more »
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
New!!: Integral domain and Manifold · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
New!!: Integral domain and Mathematics · See more »
Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
New!!: Integral domain and Matrix (mathematics) · See more »
Matrix ring
In abstract algebra, a matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication.
New!!: Integral domain and Matrix ring · See more »
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.
New!!: Integral domain and Minimal prime ideal · See more »
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.
New!!: Integral domain and Monoid · See more »
Nilradical of a ring
In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements of the ring.
New!!: Integral domain and Nilradical of a ring · See more »
Open set
In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.
New!!: Integral domain and Open set · See more »
P-adic number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.
New!!: Integral domain and P-adic number · See more »
Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
New!!: Integral domain and Polynomial · See more »
Primary decomposition
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals).
New!!: Integral domain and Primary decomposition · See more »
Prime element
In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials.
New!!: Integral domain and Prime element · See more »
Prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.
New!!: Integral domain and Prime ideal · See more »
Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
New!!: Integral domain and Prime number · See more »
Principal ideal
In the mathematical field of ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x of P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept.
New!!: Integral domain and Principal ideal · See more »
Product ring
In mathematics, it is possible to combine several rings into one large product ring.
New!!: Integral domain and Product ring · See more »
Quadratic integer
In number theory, quadratic integers are a generalization of the integers to quadratic fields.
New!!: Integral domain and Quadratic integer · See more »
Quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.
New!!: Integral domain and Quotient ring · See more »
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.
New!!: Integral domain and Rational number · See more »
Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
New!!: Integral domain and Real number · See more »
Reduced ring
In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements.
New!!: Integral domain and Reduced ring · See more »
Regular local ring
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.
New!!: Integral domain and Regular local ring · See more »
Ring of integers
In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in.
New!!: Integral domain and Ring of integers · See more »
Serge Lang
Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician and activist.
New!!: Integral domain and Serge Lang · See more »
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by \operatorname(R), is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
New!!: Integral domain and Spectrum of a ring · See more »
Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
New!!: Integral domain and Springer Science+Business Media · See more »
Subclass (set theory)
In set theory and its applications throughout mathematics, a subclass is a class contained in some other class in the same way that a subset is a set contained in some other set.
New!!: Integral domain and Subclass (set theory) · See more »
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).
New!!: Integral domain and Subring · See more »
Tensor product of algebras
In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra.
New!!: Integral domain and Tensor product of algebras · See more »
Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
New!!: Integral domain and Topological space · See more »
Unique factorization domain
In mathematics, a unique factorization domain (UFD) is an integral domain (a non-zero commutative ring in which the product of non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers.
New!!: Integral domain and Unique factorization domain · See more »
Unit (ring theory)
In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.
New!!: Integral domain and Unit (ring theory) · See more »
Unit interval
In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1.
New!!: Integral domain and Unit interval · See more »
Wedderburn's little theorem
In mathematics, Wedderburn's little theorem states that every finite domain is a field.
New!!: Integral domain and Wedderburn's little theorem · See more »
Zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that, or equivalently if the map from to that sends to is not injective.
New!!: Integral domain and Zero divisor · See more »
Zero element
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures.
New!!: Integral domain and Zero element · See more »
Zero ring
In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element.
New!!: Integral domain and Zero ring · See more »
Zero-product property
In algebra, the zero-product property states that the product of two nonzero elements is nonzero.
New!!: Integral domain and Zero-product property · See more »
1
1 (one, also called unit, unity, and (multiplicative) identity) is a number, numeral, and glyph.
New!!: Integral domain and 1 · See more »
Redirects here:
Associate (ring theory), Associate elements, Associated element, Associatedness, Cancellation ring, Divisibility in rings, Entire ring, Integral domains, Integral ring.
References
[1] https://en.wikipedia.org/wiki/Integral_domain
