Irreducible element

In a commutative unital ring
A nonzero element in a commutative unital ring is said to be irreducible if it is neither zero nor a unit, and given any factorization of the element as a product of two elements of the ring, it is associate to one of them.

In an integral domain
In an integral domain, there are two equivalent formulations. A nonzero element in an integral domain is said to be irreducible if it is neither zero nor a unit, and it satisfies the following equivalent conditions:


 * Any expression of it as a product of two elements has the property that one of the factors is associate to it.
 * Any expression of it as a product of two elements has the property that one of the factors is a unit.

Stronger properties

 * Weaker than::Prime element:
 * Weaker than::Universal side divisor: