Letting $X$ be a ring and $K$ be an $X$-module, I need to show that if $K \cong A \times B$ for some $X$-modules $A,B$, then $\exists$ submodules $M'$ and $N'$ of $K$ such that:
$K=M' \oplus N'$
$M' \cong A$
$N' \cong B.$
I understand the concepts of internal and external direct sum of modules, and I showed that if $K = M \oplus N$ for $M,N$ submodules of $K$, then $K \cong M \times N.$ (I showed the isomorphism by defining a well-defined map, and then showing that the map is a surjective homomorphism, followed by the kernel being $\{0\}$ and applying the First Isomorphism Theorem.)
But I have tried doing this problem for hours now, and have not been able to crack it. How should I begin?
Hint:
Let $\phi : A\times B\to K$ be the isomorphism.
Look at the submodules of $K$ which are given by $\phi(A\times \{0\})$ and $\phi(\{0\}\times B)$.