In algebra, an irreducible element of an integral domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.
If the irreducible factors of every non-zero non-unit element are uniquely defined, up to the multiplication by a unit, then the integral domain is called a unique factorization domain, but this does not need to happen in general for every integral domain.
It was discovered in the 19th century that the rings of integers of some number fields are not unique factorization domains, and, therefore, that some irreducible elements can appear in some factorization of an element and not in other factorizations of the same element.
The ignorance of this fact is the main error in many of the wrong proofs of Fermat's Last Theorem that were given during the three centuries between Fermat's statement and Wiles's proof of Fermat's Last Theorem.
implies that the ideal generated by
is equal to the ideal generated by
This equivalence does not hold for general commutative rings, which is why the assumption of the ring having no nonzero zero divisors is commonly made in the definition of irreducible elements.
It results also that there are several ways to extend the definition of an irreducible element to an arbitrary commutative ring.
(A non-zero non-unit element
) In an integral domain, every prime element is irreducible,[a][2] but the converse is not true in general.
The converse is true for unique factorization domains[2] (or, more generally, GCD domains).
is prime, and so the ideal generated by
is a prime (hence irreducible) ideal of
In the quadratic integer ring
it can be shown using norm arguments that the number 3 is irreducible.
However, it is not a prime element in this ring since, for example, but 3 does not divide either of the two factors.