More general. Let $\{I_j:j\in J\}$ be a family of ideals of a ring $A$. Are $\displaystyle\sum_{j\in J}I_j$ and $\displaystyle\bigcap_{j\in J}I_j$ ideals of $A$?
since \(\displaystyle I,J\) are both ideals of \(\displaystyle A\) (and thus additive subgroups).
This shows \(\displaystyle I+J\) is an additive subgroup of \(\displaystyle (A,+)\).
Now let \(\displaystyle a \in A\) be any element. We have:
\(\displaystyle ar = a(x + y) = ax + ay \in I + J\), because \(\displaystyle I,J\) are both IDEALS.
The proof that \(\displaystyle ra \in I + J\) is similar, and left to the reader.
This proof clearly generalizes to any family of ideals indexed by a FINITE set. The infinite case has some complications better off discussed elsewhere.
A similar approach works for \(\displaystyle I \cap J\). It should be clear that \(\displaystyle I \cap J\) is an additive subgroup of \(\displaystyle A\). I hope you can see how to prove that for any:
\(\displaystyle a \in A, r \in I \cap J\) that \(\displaystyle ar,ra \in I \cap J\).