Irreducible element

Definition

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.

Relation with other properties

Stronger properties

• Prime element: For proof of the implication, refer Prime element implies irreducible and for proof of its strictness (i.e. the reverse implication being false) refer Irreducible element not implies prime
• Universal side divisor: For proof of the implication, refer Universal side divisor implies irreducible and for proof of its strictness (i.e. the reverse implication being false) refer Irreducible not implies universal side divisor