In an integral domain , the decomposition of a nonzero noninvertible element as a product of prime (or irreducible) factors
(1)
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is unique if every other decomposition of the same type has the same number of factors
(2)
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and its factors can be rearranged in such a way that for all indices , and differ by an invertible factor.
The prime factorization of an element, if it exists, is always unique, but this does not apply, in general, to irreducible factorizations: in the ring ,
(3)
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are two different irreducible factorizations, none of which is prime. 2 is not a prime element in , since it does not divide either of the factors of the middle expression. In fact
(4)
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lie both outside . Furthermore,
(5)
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which shows that is not prime either.
An integral domain where every nonzero noninvertible element admits a unique irreducible factorization is called a unique factorization domain.