## The set of ideals of a ring is closed under arbitrary intersections

Let $R$ be a ring.

1. Prove that if $I$ and $J$ are ideals of $R$, then $I \cap J$ is also an ideal of $R$.
2. Prove that if $I_k$ is an ideal of $R$ for each $k \in A$, $A$ an arbitrary set, then $\bigcap_A I_k$ is also an ideal of $R$.

1. In this previous exercise, we showed that $I \cap J$ is a subring of $R$. Thus it suffices to show that $I \cap J$ absorbs $R$ on the right and the left. To that end, let $r \in R$ and $x \in I \cap J$. Now $x \in I$, so that $rx, xr \in I$. Likewise, $rx, xr \in J$. Thus $rx, xr \in I \cap J$, and $I \cap J \subseteq R$ is an ideal.
2. Again, we showed in this previous exercise that $\bigcap_A I_k$ is a subring of $R$, os it suffices to show absorption. Let $r \in R$ and $x \in \bigcap_A I_k$. Now $x \in I_k$ for each $k \in A$, so that $rx, xr \in I_k$ for all $k \in A$. Thus $xr, rx \in \bigcap_A I_k$. Hence $\bigcap_A I_k$ is an ideal of $R$.