Irreducible element

In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units, or equivalently, if every factoring of such element contains at least one unit.
Relationship with prime elements.
Irreducible elements should not be confused with prime elements. (A non-zero non-unit element formula_1 in a commutative ring formula_2 is called prime if, whenever formula_3 for some formula_4 and formula_5 in formula_6 then formula_7 or formula_8) In an integral domain, every prime element is irreducible, but the converse is not true in general. The converse is true for unique factorization domains (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. However, if formula_9 is a GCD domain and formula_10 is an irreducible element of formula_9, then as noted above formula_10 is prime, and so the ideal generated by formula_10 is a prime (hence irreducible) ideal of formula_9.
Example.
In the quadratic integer ring formula_15 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.