# Irreducible element

In ring theory a element of a ring is said to be **irreducible** if:

- is not a unit.
- cannot be written as the product of two non-units in , that is if for some then either or is a unit in .

This is analogous to the definition of prime numbers in the integers and indeed in the ring the irreducible elements are precisely the primes numbers and their negatives.

In a principal ideal domain it is easy to see that the ideal is maximal iff is irreducible. Indeed, we have iff so if is irreducible then or (since , either is a unit (so ) or is times a unit (so )). Conversely if is maximal then if we have so hence either or . In the first case is a unit and in the second case , where is a unit, and hence , a unit. So in either case is irreducible.

*This article is a stub. Help us out by expanding it.*