Let $M$ be an $R$-module and $A$ and $B$ submodules of $M$ such that $M=A\oplus B$.
If $C$ and $D$ are submodules of $A$ and $B$, respectively and $N=C+D$, then prove that:
$$N=C\oplus D \text{ and } \frac{M}{N}=\frac{A\oplus B}{C \oplus D}\cong \frac{A}{C} \oplus \frac{B}{D}.$$
Is there an extension to this question? ( for part 2 )
On
In general, if $f_i:M_i\rightarrow N_i$ is a homomorphism of R-modules for each $i\in I$, the homomorphism $$\begin{matrix}f:&\displaystyle\bigoplus_{i\in I}M_i&\longrightarrow&\displaystyle\bigoplus_{i\in I}N_i\\&(x_i)_{i\in I}&\longmapsto&(f_i(x_i))_{i\in I}\end{matrix}$$ is such that $\displaystyle Ker(f)=\bigoplus_{i\in I}Ker(f_i)$ and $Im(f)=\displaystyle\bigoplus_{i\in I}Im(f_i)$. In particular, when $N_i$ is a submodule of $M_i$ and $f_i$ is the canonical projection $M_i\rightarrow\frac{M_i}{N_i}$ then $f$ is surjective and $$\frac{\displaystyle\oplus_{i\in I}M_i}{\displaystyle\oplus_{i\in I}N_i}=\frac{\oplus_{i\in I}M_i}{\oplus_{i\in I}Ker(f_i)}=\frac{\oplus_{i\in I}M_i}{Ker(f)}\simeq\bigoplus_{i\in I}\frac{M_i}{N_i}.$$
I'm not completely sure what you meant to ask, but:
1) Clearly $\;C\cap D=\{0\}\;$ , and then $\;C+D=C\oplus D\;$
(2) First, observe that $\; A/C\oplus B/D\;$ is the external direct sum, and now define
$$\phi: M=A\oplus B\to A/C\oplus B/D\;,\;\;\phi(a,b):=\left(a+C\,,\,\,b+D\right)$$
Prove $\;\phi\;$ is an $\;R\,-$ modules homomorphism, and now find its kernel. Finally, use the first isomorphism theorem.