----- > [!proposition] Proposition. ([[ascending chain condition for existence of factorization]]) > > Let $R$ be an [[integral domain]], and let $r$ be a nonzero, nonunit element of $R$. Assume that the [[principal ideal|principal ideals]] of $R$ satisfy an [[ascending chain condition]]: every ascending chain of principal ideals $\langle r \rangle \subset \langle r_{1} \rangle \subset \langle r_{2} \rangle \subset \langle r_{3} \rangle \subset \cdots $ > stabilizes. Then $r$ has a [[factorization into irreducibles]]. > > Thus, factorizations exist in integral domains in which the [[ascending chain condition]] holds *for principal ideals*. > [!proposition] Corollary. > Let $R$ be a [[Noetherian domain]]. Then factorizations exist in $R$. > > (Indeed, [[ascending chain - maximality characterization of Noetherian modules]] says that Noetherian domains satisfy the [[ascending chain condition]] for *all* [[ideal|ideals]].) ^proposition > [!proof]- Proof. ([[ascending chain condition for existence of factorization]]) > Contrapositively assume $r$ does *not* have a [[factorization into irreducibles]]. We will construct an infinite ascending chain as follows. > First note that, in particular, $r$ is not [[irreducible element of an integral domain|irreducible]]. So $\langle r \rangle$ is not maximal among proper [[principal ideal|principal ideals]]; this lets us write $r=r_{1}s_{1}$ for some $r_{1},s_{1} \in R$ with $\langle r \rangle \subsetneq \langle r_{1} \rangle$ and $\langle r \rangle \subsetneq \langle s_{1} \rangle$. If both $r_{1},s_{1}$ factor into irreducibles then $r$ [[factorization into irreducibles|factors into irreducibles]] via the product of these factorizations; so assume (e.g.) $r_{1}$does not have a [[factorization into irreducibles]]. But then we can get $r_{1}=r_{2}s_{2}$ for $\langle r_{1} \rangle \subsetneq \langle r_{2} \rangle$ and $\langle r_{1} \rangle \subsetneq \langle s_{2} \rangle$... iterating the argument gives the infinitely increasing chain $\langle r \rangle \subsetneq \langle r_{1} \rangle \subsetneq \langle r_{2} \rangle \subsetneq\langle r_{3} \rangle \subsetneq \cdots$ that we'd hoped to obtain. ----- #### ---- #### 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 > ```