nLab irreducible element

Contents

Context

Algebra

Monoid theory

Contents

Definition

In a monoid, an element xx is irreducible if it is neither invertible nor the product of two non-invertible elements. Without bias, we can say that xx 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

Last revised on January 22, 2023 at 20:40:20. See the history of this page for a list of all contributions to it.