Let X, Y , and Z be $R$- Modules, for $R$ a commutative ring. Furthermore, let $X$ and $Y$ be submodules of $Z$. I want to show that if $X + Y = Z$ and $X \cap Y = \{0 \}$, then $Z \cong X \oplus Y$.
To start, I am defining a map $\phi$: $X \oplus Y \rightarrow Z$ by $(x,y) \mapsto x + y$.
My question is, will this map do it for me? In otherwords, it is possible to prove that $\phi$ is an isomorphism of $R$ modules, or am I going down the wrong path with this mapping. Thank you.
Just use your proposed map. To see surjectivity, consider any $z \in Z$. Since $Z = X + Y$, there exists $x \in X$ and $y \in Y$ such that $z = x + y$. Now $\phi(x, y) = z$, proving surjectivity.
To see injectivity, suppose $\phi(x_1, y_1) = \phi(x_2, y_2)$. Then $x_1 + y_1 = x_2 + y_2$. This implies $x_1 - x_2 = y_2 - y_1$. Since both $X, Y$ are submodules, we see that the left-hand side is in $X$ and the right-hand side is in $Y$. Thus they are both in the intersection $X \cap Y = \{0\}$. So $x_1 - x_2 = y_2 - y_1 = 0$, i.e. $(x_1, x_2) = (y_1, y_2)$, proving injectivity.
It is also easy to see that this is a $R$-linear map. $$\phi(rx, ry) = rx+ry = r(x+y) = r\phi(x, y).$$
So your map is indeed an $R$-module isomorphism.