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Let $R$ be an arbitrary ring and $M$ be a right $R$-module with $M=A \oplus B$.

Suppose that C is a submodule of $M$ such that $C \cap A = 0$. Can we say that $C \subseteq B$ ?

I think, it is not always true, but I can not find a counter example.

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It is not true. Take for instance $R=\mathbb{Z}$, $A=B=\mathbb{Z}$ and $C=(1,1)\mathbb{Z}$. Since $A$ in $M=A\oplus B$ consists of the elements $(n,0)$ for $n\in\mathbb{Z}$, $A\cap C=0$, yet clearly $C$ is not contained in $B$. In fact, $C\cap B=0$ as well.

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