As this is a coursework question I would appreciate it if any answers only give hints. So if $A, B, C$ are $R$-modules (for some ring $R$) such that $A\cap B=\{0\}$. I am tasked with deciding whether $A\oplus B\cong A\oplus C$ if $B\cong C$.
So far I was thinking of letting $f:B\cong C$ be the isomorphism given, and extending this to a map $f_\ast:A\oplus B\to A+C$ by $f_\ast(a,b)=a+f(b)$. It is obvious that this is well defined, and it is easy to show $f_\ast$ is an $R$-module homomorphism. Likewise, it is easy to see that $f_\ast$ is surjective. My trouble is showing this is injective. Here is what I have so far:
$f_\ast(a,b)=0$ if $a=-f(b)$. So $a\in A\cap C$. Does $A\cap C=\{0\}$?
Edit: I think I am only able to assume $A\cap B=\{0\}$. So that I need to show $A\cap C=\{0\}$. Although, can I not define a map $g_\ast:A\cap C\to A\cap B$ by $g_\ast(x)=g(x)$, where $g$ is the inverse of $f$. This should be a homomorphism and is obviously surjective because $A\cap B=\{0\}$.
Couple things I need to point out. First of all, $A\cap B$, $A\cap C$, and $A+C$ only make sense in the context where $A,B,C$ are all submodules of a larger submodule.
What does $A\cap B$ mean? Well, it's only necessarily an $R$-module when $A$ and $B$ are both submodules of the same $R$-module $M$. So it's not really defined when $A$ and $B$ are general $R$-modules.
What is $A+C$? This one is certainly only defined when $A$ and $B$ are both submodules of a larger module $M$, since it's defined as one of the two equivalent definitions:
Thus it doesn't really make sense when you say $A+C$, $A\cap B$, and $A\cap C$. This takes me to the next thing I need to point out.
Second thing, there are two types of direct sums usually discussed in introductory classes, internal and external direct sums.
We say that $M$ is an internal direct sum of submodules $A$ and $B$ if $A+B=M$ and $A\cap B =0$.
On the other hand, we define the external direct sum, or just direct sum of modules $A$ and $B$ to be the set $A\times B$ with pointwise product and sum. For clarity, I will only use $A\oplus B$ for the external direct sum, though in practice its often used when $A$ and $B$ are submodules for the internal direct sum as well. In this case, we usually identify $A$ and $B$ with the submodules $A\times \{0\}$ and $\{0\}\times B$ respectively.
It turns out that $A\oplus B$ is the internal direct sum of $A$ and $B$, and if $M$ is the internal direct sum of $A$ and $B$, then it is isomorphic to $A\oplus B$, so these are more or less equivalent.
The reason I bring this up however, is that based on your usage of $A\cap B$, $A\cap C$ and $A+C$ in your question, it appears that you're thinking about the internal direct sum definition rather than the external direct sum, which is what appears to be intended based on the question as stated, which doesn't seem to state or imply that $A$, $B$, and $C$ are submodules of some module. However you also used the external direct sum definition of $A\oplus B$, which leaves me confused.
Thus you should probably define $F$ as in freakish's answer by $F(a,b)=(a,f(b))$. Alternatively, you can identify $A$ and $C$ with submodules of $A\oplus C$ and proceed as in your question, but then since $A\oplus C$ is the internal direct sum of $A$ and $C$, $A\cap C=0$ by definition, as you needed.