Irreducible element

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Template:Associate-invariant curing-element property

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