nLab unique factorization monoid

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.

Related concepts

• fundamental theorem of arithmetic

• ideal of a monoid

• principal ideal of a monoid

• principal ideal monoid