---- > [!definition] Definition. ([[UFD]]) > An [[integral domain]] $R$ is **factorial**, or a **unique factorization domain (UFD)**, if it is a [[domain with factorizations|domain with factorizations (into irreducibles)]] wherein every [[factorization into irreducibles|factorization]] is *unique*. ^definition > [!basicproperties] > Let $R$ be a UFD, and let $a,b,c$ be nonzero elements of $R$. Then > - $\langle a \rangle \subset \langle b \rangle$ iff the multiset of [[irreducible element of an integral domain|irreducible]] factors of $b$ is contained in the multiset of irreducible factors of $a$; > - $a,b$ are [[divides|associates]] (i.e., $\langle a \rangle=\langle b \rangle$) iff the two multisets coincide; > - The [[irreducible element of an integral domain|irreducible]] factors of a product $bc$ are the collection of all [[irreducible element of an integral domain|irreducible factors]] of $b$ and of $c$. ^properties > [!proof] > **1.** > > $\to$. Suppose $\langle a \rangle \subset \langle b \rangle$. Write $a=p_{1}\cdots p_{n}$ and $b=q_{1} \cdots q_{m}$. $a=bc$ for some $c \in R$, i.e., $a=q_{1} \cdots q_{m} c$ for irreducible elements $q_{1},\dots,q_{m}$. Uniqueness of factorizations then implies each $q_{i}$ equals some $p_{j}$ (up to association?), proving this direction. > > $\leftarrow$. Suppose the multiset of irreducible factors of $b$ is contained in that of $a$. Then $a=q_{1}\dots q_{m}c=bc$ for some $c$ and this suffices to conclude $\langle a \rangle \subset \langle b \rangle$. > > **2.** If $a,b$ are [[divides|associates]] then in particular $\langle a \rangle \subset \langle b \rangle$, yielding by the above one direction of inclusion. But also $\langle b \rangle \subset \langle a \rangle$, yielding the reverse inclusion. Conversely, if the two multisets coincide then uniqueness immediately enforces that $a,b$ are equal up to association. > > **3.** With $b=q_{1} \cdots q_{m}$ and $c=k_{1} \cdots k_{\ell}$, $bc=q_{1} \cdots q_{m} k_{1} \cdots k_{\ell}$ and so we have found one factorization of the product into irreducibles; by uniqueness it is the only one. ---- #### ---- #### 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 > ```