---- > [!definition] Definition. ([[ideal quotient]]) > Let $I$ and $J$ be [[ideal|ideals]] of a [[commutative ring|commutative]] [[ring]] $R$. Their **ideal quotient** is the [[ideal]] $(I: J)=\{ r \in R: rJ \subset I \}.$ > Since for any $r, x \in R$, $r\langle x \rangle \subset I \iff rx \in I$, the notation $(I:x)$ is used to mean $(I: \langle x \rangle)$ . > [!justification] > Need to check this is indeed an [[ideal]]. Let $s \in R$, $r \in (I:J)$. Then $(sr)J=s\underbrace{(rJ)}_{\subset I}$ and since $I$ is an [[ideal]] this means the entire product is contained in $I$. ^justification ---- #### ---- #### 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 > ```