nLab irreducible element

Definition

In a monoid, an element $x$ is irreducible if it is neither invertible nor the product of two non-invertible elements. Without bias, we can say that $x$ is irreducible if, whenever it is written as a product of a finite list of elements, at least one element in the list is invertible.

In a commutative ring, an element is irreducible if it is neither invertible nor the product of two non-invertible elements, with respect to the multiplication operation on the commutative ring.

Examples

• Every prime number is an irreducible element in the integers.

• Given a field $K$, every monic polynomial of degree one is an irreducible element in the univariate polynomial ring $K[x]$.

Related concepts

• unique factorization monoid
• unique factorization domain