nLab
unique factorization monoid
Contents
Context
Algebra
Monoid theory
monoid theory in algebra :

monoid , infinity-monoid

monoid object , monoid object in an (infinity,1)-category

Mon , CMon

monoid homomorphism

trivial monoid

submonoid , quotient monoid?

divisor , multiple? , quotient element?

inverse element , unit , irreducible element

ideal in a monoid

principal ideal in a monoid

commutative monoid

cancellative monoid

GCD monoid

unique factorization monoid

Bézout monoid

principal ideal monoid

group , abelian group

absorption monoid

free monoid , free commutative monoid

graphic monoid

monoid action

module over a monoid

localization of a monoid

group completion

endomorphism monoid

super commutative monoid

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 22:26:58.
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