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35 relations: Abelian group, Algebraic integer, Ascending chain condition on principal ideals, Atomic domain, Étienne Bézout, Bézout's identity, Dedekind domain, Entire function, Field of fractions, Finitely generated module, Free ideal ring, French people, GCD domain, Greatest common divisor, Hereditary ring, Integral domain, Irreducible element, Irving Kaplansky, Linear combination, Local ring, Localization of a ring, Mathematician, Mathematics, Maximal ideal, Noetherian ring, Ore condition, Prüfer domain, Prime element, Prime ideal, Principal ideal, Principal ideal domain, Principal ideal ring, Total order, Unique factorization domain, Valuation ring.
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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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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Ascending chain condition on principal ideals
In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion.
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Atomic domain
In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain, every non-zero non-unit of which can be written (in at least one way) as a (finite) product of irreducible elements.
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Étienne Bézout
Étienne Bézout (31 March 1730 – 27 September 1783) was a French mathematician who was born in Nemours, Seine-et-Marne, France, and died in Avon (near Fontainebleau), France.
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Bézout's identity
In elementary number theory, Bézout's identity (also called Bézout's lemma) is the following theorem: The integers x and y are called Bézout coefficients for (a, b); they are not unique.
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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.
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Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.
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Field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded.
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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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Free ideal ring
In mathematics, especially in the field of ring theory, a (right) free ideal ring, or fir, is a ring in which all right ideals are free modules with unique rank.
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French people
The French (Français) are a Latin European ethnic group and nation who are identified with the country of France.
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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).
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Greatest common divisor
In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.
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Hereditary ring
In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective.
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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.
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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.
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Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and musician.
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Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).
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Local ring
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.
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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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Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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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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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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Ore condition
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring.
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Prüfer domain
In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context.
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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.
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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.
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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.
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Principal ideal domain
In abstract algebra, 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.
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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 one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring.
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Total order
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.
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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.
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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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References
[1] https://en.wikipedia.org/wiki/Bézout_domain
