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.

Relationship with prime elements

Irreducible elements should not be confused with prime elements. 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
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 is a GCD domain and is an irreducible element of, then as noted above is prime, and so the ideal generated by is a prime ideal of.

Example

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.