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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 ...
There is a version of unique factorization for ordinals, though it requires some additional conditions to ensure uniqueness. Any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers.
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 class number of a number field is by definition the order of the ideal class group of its ring of integers. Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q has class number 1.
As for every unique factorization domain, every Gaussian integer may be factored as a product of a unit and Gaussian primes, and this factorization is unique up to the order of the factors, and the replacement of any prime by any of its associates (together with a corresponding change of the unit factor).
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 Bézout domain (i.e., an integral domain where every finitely generated ideal is principal) is a GCD domain. Unlike principal ideal domains (where every ideal is principal), a Bézout domain need not be a unique factorization domain; for instance the ring of entire functions is a non-atomic Bézout domain, and there are many other
An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID.