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, all but one elements in the list are 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.