# 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 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 does not divide either of the two factors.^{[4]}

## See also

## References

- ↑ Consider a prime that is reducible: Then or Say then we have Because is an integral domain we have So is a unit and is irreducible.
- 1 2 Sharpe (1987) p.54
- ↑ http://planetmath.org/encyclopedia/IrreducibleIdeal.html
- ↑ 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.