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**Bernhard** Proof

$\displaystyle x, y \in \displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha} $

$\displaystyle \longrightarrow $ $\displaystyle x, y \in I_\alpha $ for all $\displaystyle \alpha \in \Lambda$

$\displaystyle \longrightarrow $ $\displaystyle x - y \in I_\alpha $ for all $\displaystyle \alpha \in \Lambda $

$\displaystyle \longrightarrow $ $\displaystyle x - y \in \displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha}$ .... (1)

$\displaystyle x \in I \cap J $

$\displaystyle \longrightarrow $ $\displaystyle x \in I_\alpha $ for all $\displaystyle \alpha \in \Lambda $

$\displaystyle \longrightarrow $ $\displaystyle rx \in I_\alpha $ for all $\displaystyle \alpha \in \Lambda $ and for all $\displaystyle r \in R $

$\displaystyle \longrightarrow $ $\displaystyle rx \in \displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha} $ ..... (1)

(1) , (2) $\displaystyle \longrightarrow $ $\displaystyle \displaystyle{\bigcap_{\alpha \in \Lambda}I_\alpha} $ is ideal of R