"Unique factorization" redirects here. For the uniqueness of integer factorization, see fundamental theorem of arithmetic.
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In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units.
Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field.
Unique factorization domains appear in the following chain of class inclusions:
example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x2 – 4. Factorization is not usually considered meaningful...
In mathematics, a noncommutative uniquefactorizationdomain is a noncommutative ring with the uniquefactorization property. The ring of Hurwitz quaternions...
called the uniquefactorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product...
Principal ideal domains are Noetherian, they are integrally closed, they are uniquefactorizationdomains and Dedekind domains. All Euclidean domains and all...
integer coefficients (or, more generally, with coefficients in a uniquefactorizationdomain) is the greatest common divisor of its coefficients. The primitive...
in uniquefactorizationdomains. The polynomial ring F[x] over a field F (or any unique-factorizationdomain) is again a uniquefactorizationdomain. Inductively...
non-associate divisors). Every uniquefactorizationdomain obviously satisfies these two conditions, but neither implies uniquefactorization. P.M. Cohn, Bezout rings...
such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that...
factorizationdomains, and, therefore, that some irreducible elements can appear in some factorization of an element and not in other factorizations of...
they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies uniquefactorization and many related properties...
hold for uniquefactorizationdomains. The fundamental theorem of arithmetic continues to hold (by definition) in uniquefactorizationdomains. An example...
divisor Principal ideal domain, an integral domain in which every ideal is principal Uniquefactorizationdomain, an integral domain in which every non-zero...
integral domains. If R is a uniquefactorizationdomain then the same holds for R[X]. This results from Gauss's lemma and the uniquefactorization property...
The proof that a polynomial ring over a uniquefactorizationdomain is also a uniquefactorizationdomain is similar, but it does not provide an algorithm...
domain: Any number from a Euclidean domain can be factored uniquely into irreducible elements. Any Euclidean domain is a uniquefactorizationdomain (UFD)...
that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors....
irreducible element (up to multiplication by units). R is a uniquefactorizationdomain with a unique irreducible element (up to multiplication by units). R...
domain R, every element can be factorized into irreducible elements (in short, R is a factorizationdomain). Thus, if, in addition, the factorization...
algorithm for integer factorization. The same method can also be illustrated with a Venn diagram as follows, with the prime factorization of each of the two...
the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems. The first polynomial factorization algorithm...
Consequently, in a Schreier domain, every irreducible is prime. In particular, an atomic Schreier domain is a uniquefactorizationdomain; this generalizes the...
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