nLab
unique factorization monoid

Contents

Context

Algebra

Monoid theory

Contents

Definition

Let MM be a monoid. We say that an element aMa\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 MM is a unique factorization monoid if every non-unit has a factorization u=m 1m nu = 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: MM is a unique factorization monoid precisely when the monoid of principal ideals of MM 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.