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Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R which is not a unit can be written as a finite product of irreducible elements p i of R: x = p 1 p 2 ⋅⋅⋅ p n with n ≥ 1. and this representation is unique in the following sense: If q 1, ..., q m are irreducible elements ...
In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property. Examples The ...
In the case of coefficients in a unique factorization domain R, "rational numbers" must be replaced by "field of fractions of R". This implies that, if R is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over R is a unique factorization domain. Another ...
A principal ideal domain (in particular: the integers and any field). A unique factorization domain (in particular, any polynomial ring over a field, over the integers, or over any unique factorization domain). A GCD domain (in particular, any Bézout domain or valuation domain). A Dedekind domain.
The converse is true for unique factorization domains [2] (or, more generally, GCD domains). Moreover, while an ideal generated by a prime element is a prime ideal , it is not true in general that an ideal generated by an irreducible element is an irreducible ideal .
The unique factorization property means that a non-zero non-unit r can be represented as a product of prime elements r = p 1 p 2 ⋯ p n {\displaystyle r=p_{1}p_{2}\cdots p_{n}} Then r is square-free if and only if the primes p i are pairwise non-associated (i.e. that it doesn't have two of the same prime as factors, which would make it ...
Examples of regular rings include fields (of dimension zero) and Dedekind domains. If A is regular then so is A [ X ], with dimension one greater than that of A . In particular if k is a field, the ring of integers, or a principal ideal domain , then the polynomial ring k [ X 1 , … , X n ] {\displaystyle k[X_{1},\ldots ,X_{n}]} is regular.
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm.