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Wikipedia gives the definition of a Unique Factorisation Domain as one where every element "can be written as a product of prime elements (or irreducible elements)" which suggests that in a UFD prime and irreducible elements are the same.

However, I thought that only in a PID were prime and irreducible elements the same and in a UFD it is true that all prime elements are irreducible, but not visa versa.

What's going on here?

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    It should say irreducible, see Lang for example. Also, "However, in unique factorization domains,[3] or more generally in GCD domains, primes and irreducibles are the same." http://en.wikipedia.org/wiki/Prime_element2012-05-14
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    Prime elements are irreducible in any domain (this is a nice exercise). In a UFD, irreducible elements are prime, and in a PID, irreducible elements (which are also prime elements) generate maximal ideals, since the ideal generated by an irreducible element is maximal among proper principal ideals.2012-05-14
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    If you have access to I. M. Isaacs' graduate algebra text, the results Tobias mentioned are right next to each other on page 240.2012-05-14

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