----- > [!proposition] Proposition. ([[sum of ideals is an ideal]]) > If $\{ I_{\alpha} \}_{\alpha \in A}$ is a family of [[ideal|ideals]] of a [[ring]] $R$, then the sum $\sum_{\alpha}I_{\alpha}$ is an [[ideal]] of $R$. > > The intersection $\bigcap_{\alpha}^{}I_{\alpha}$ is an [[ideal]]. ^proposition > [!proof]- Proof. ([[sum of ideals is an ideal]]) > The general element of $\sum_{\alpha}I_{\alpha}$ looks like a sum $\sum_{\alpha}i_{\alpha}$ where each $i_{\alpha} \in I_{\alpha}$. Thus the general element of $r \sum_{\alpha}I_{\alpha}$ looks like $r\sum_{\alpha}i_{\alpha}$, which we may distribute over to obtain $\sum_{\alpha}ri_{\alpha}$. Since each $I_{\alpha}$ is an [[ideal]], $ri_{\alpha} \in I_{\alpha}$. Hence their sum belongs to $\sum_{\alpha}i_{\alpha}$. Checking right-ideal is similar. > >Let $i$ in the intersection, so that $i \in I_{\alpha}$ for each $\alpha$. THen $ri \in I_{\alpha}$ for each $\alpha$, hence $ri$ lives in the intersection too. ----- #### ---- #### 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 > ```