---- > [!definition] Definition. ([[factorization into irreducibles]]) > Let $R$ be a (usually [[commutative ring|commutative]]) [[integral domain]]. An element $r \in R$ has a **factorization (or decomposition) into irreducibles** if there exist [[irreducible element of an integral domain|irreducible elements]] $q_{1},\dots,q_{n}$ such that $r=q_{1} \cdots q_{n}$. > > We call such a factorization **unique** if the $q_{i}$ are determined up to order and [[divides|associates]], that is, if whenever $r=q_{1}' \cdots q_{m}'$ > is another factorization of $r$ into [[irreducible element of an integral domain|irreducibles]], then $m=n$ and $q_{i}'$ is an associate of $q_{i}$ after (possibly) shuffling up the factors. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```