Bézout domain and Unique factorization domain

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Difference between Bézout domain and Unique factorization domain

Bézout domain vs. Unique factorization domain

In mathematics, a Bézout domain is a form of a Prüfer 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.

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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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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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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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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Localization of a ring

In commutative algebra, localization is a systematic method of adding multiplicative inverses to a ring.

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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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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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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 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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