Prove these subrings are ideals Let $R = R_1⊕R_2$ and let $I_1$ and $I_2$ be ideals of rings $R_1$ and $R_2$ respectively. We consider $R_1$ and $R_2$ as subrings of R under canonical embeddings $R_i \rightarrow R$ and thus $I_1$ and $I_2$ are considered as subrings of R.

(a) Prove that $I_1$ and $I_2$ are ideals of $R$.
(b) If each $I_i$ is a maximal ideal of $R_i$, is the ideal $I_1 +I_2$ of $R$ maximal? Prime?

I honestly don't know how to approach either.  I think for a I want to show it just by using the definition, but that seems kind of tedious.  Is there a more efficient way?
For b, I know that an ideal I is maximal if and only if R/I is a field and an ideal P is prime if and only if R/P is an integral domain
The set $I_1\oplus 0$ is an ideal since if $x\oplus y$ is any element of $R$ and $a\oplus 0$ any element of the set, then 
$$(x\oplus y)(a\oplus 0)=xa\oplus 0\in I_1\oplus 0$$

However $I_1\oplus I_2$ need not be prime for example let $R_1=R_2=\mathbb{Z}$ and 
$$(2\oplus 1)(1\oplus 3)\in (2)\oplus (3)$$ but $2\oplus 1\not \in (2)\oplus 0$ and similarly.
For part (b), the sum of two prime ideals is not necessarily prime. Consider $\mathbb{Z}$ with the ideals $\langle 2 \rangle, \langle 3 \rangle$.

As for maximality, is $I_1 + I_2$ an ideal? $I_1$ and $I_2$ are maximal, so what can you say about $I_1 + I_2$?