# 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. (A non-zero non-unit element in a commutative ring is called prime if, whenever for some and in then or ) In an integral domain, every prime element is irreducible,[1][2] but the converse is not true in general. The converse is true for unique factorization domains[2] (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 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 .[3]

## 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.[4]

## See also

## References

- Consider a prime element of and suppose Then or Say then we have Because is an integral domain we have So is a unit and is irreducible.
- Sharpe (1987) p.54
- "Archived copy". Archived from the original on 2010-06-20. Retrieved 2009-03-18.CS1 maint: archived copy as title (link)
- William W. Adams and Larry Joel Goldstein (1976),
*Introduction to Number Theory*, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9

- Sharpe, David (1987).
*Rings and factorization*. Cambridge University Press. ISBN 0-521-33718-6. Zbl 0674.13008.