For modules: If $A \leq C$ and $A\oplus B = C\oplus B\implies A=C$?

Question

The question is pretty much in the title.

So if we are working with modules, with $A,B,C$ submodules of some module $M$ and $A\leq C$, then $M=A\oplus B = C \oplus B \implies A = C$ ?

Now, I think it makes sense that $A \cong C$ but I was wondering if we could say it was equal?

Answer 1

Let $x\in C$. Since $M=A\oplus B$, there are unique $a\in A,\ b\in B$ such that $x=a+b$. But then $C\ni x-a=b\in B$. Since $M=C\oplus B$, it must hold $x-a=0$. So $x\in A$.

Added: As you can see, the hypothesis could be weakened to $$\begin{cases} A\le C\\ C\le A+B\\ C\cap B=\{0\}\end{cases}$$