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]
Examples
The following are examples of irreducible elements:
- Irreducible polynomials.
- In the quadratic integer ring
![\mathbf{Z}[\sqrt{-5}]](http://wpcontent.answers.com/math/d/6/f/d6f15f5fb2403507e3e6c34ac4f4e648.png)
, the number 3 is irreducible but is not a prime since 9 can be written as 
and 3(3)[3].
Notes
- ^ So, in particular, not every prime ideal is irreducible in general.
- ^ http://planetmath.org/encyclopedia/IrreducibleIdeal.html
- ^ See Polcino (1972), exercise 3, p. 76.
References
- Polcino Milies, Francisco Cesar. Anéis e módulos. São Paulo. 1972.
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