Irreducible element

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In mathematics, a non-unit in an integral domain is said to be irreducible if it is not a product of two non-units. Equivalently, a non-unit x is irreducible if x ≠ 0 and every divisor d of x is associated to either 1 or x. Note this is the usual definition of a prime number.

Every prime element is irreducible. The converse is true for UFDs (or, more generally, GCD domains.)

A ideal generated by a prime element is a prime ideal. However, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal.^[1] This is the case if A is a GCD domain (in particular a UFD).^[2]

[edit] Examples

The following are examples of irreducible elements:

Irreducible polynomials.
In the quadratic integer ring $\mathbf{Z}[\sqrt{-5}]$ , the number 3 is irreducible but is not a prime since 9 can be written as $\left(2 + \sqrt{-5}\right)\left(2 - \sqrt{-5}\right)$ and $3(3)$ .