Suppose that $M, N, P$ are right $R$-modules, where $R$ is a ring with unity, such that $M=N\oplus P$. Suppose that there exist two non-zero submodules $A,B\subseteq M$ with $A \cap B = 0$ and $A\cong B$. Is it possible to construct two non-zero submodules $A^{\prime}, B^{\prime} \subseteq N$ with $A^{\prime} \cap B^{\prime} = 0$ and $A^{\prime} \cong B^{\prime}$ ?!. I thought that $\pi_N(A)$ and $\pi_N(B)$, where $\pi_N: N\oplus P \to N$ is the canonical projection, would be the target submodules but, unfortunately, they do not satisfy the required criteria.
Thanks in advance. I appreciate any help.