# nLab unique factorization monoid

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoid theory

monoid theory in algebra:

# Contents

## Definition

Let $M$ be a monoid. We say that an element $a\in M$ is a unit if it is invertible. A non-unit is called irreducible if it can not be represented as a product of two non-units.

A commutative monoid $M$ is a unique factorization monoid if every non-unit has a factorization $u = m_1 \cdots m_n$ as a product of irreducible non-units and this decomposition is unique up to renumbering and rescaling the irreducibles by units.

Put differently: $M$ is a unique factorization monoid precisely when the monoid of principal ideals of $M$ is a commutative monoid freely generated by irreducible principal ideals.

## Examples

Every abelian group is trivially a unique factorization monoid.

A unique factorization monoid object in the category of commutative semigroups is a unique factorization semiring. The statement that the semiring of positive integers is a unique factorization semiring is the fundamental theorem of arithmetic.

Last revised on May 21, 2021 at 18:26:58. See the history of this page for a list of all contributions to it.