----- > [!proposition] Proposition. ([[product of coprime ideals is their intersection]]) > Let $I_{1},\dots,I_{k}$ be [[ideal|ideals]] of a [[ring]] $R$ that are [[relatively prime integers|coprime]]: $I_{i}+I_{j}=(1)$ for all $i \neq j$. Then $I_{1} \cdots I_{k} =I_{1} \cap \dots \cap I_{k}$. ^proposition > [!proof]- Proof. ([[product of coprime ideals is their intersection]]) > By a lemma in [[Chinese remainder theorem]], we have $I_{1} \cdots I_{k-1}+I_{k}=(1)$ for $k \geq 3$. Thus the general statement is reduced by induction to the case $k=2$. Assume $I$ and $J$ are [[ideal|ideals]] of $R$ such that $I+J=(1)$. The inclusion $IJ \subset I \cap J$ holds for all ideals $I,J$, so the task amounts to proving $I \cap J \subset IJ$ when $I+J = (1)$. If $I+J=(1)$, then there exist elements $a\in I$, $b \in J$, such that $a+b=1$. But if $r \in I\cap J$. Then multiply it through the identity $a+b=1$: $ra+rb=r$. We have just written $r$ as a linear combination of products of pairs of elements in $I$ and $J$, i.e., as an element in $IJ$. ----- #### ---- #### 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 > ```