number of submodules of direct sum of simple modules

Question

Let $M$ be a simple $R$ module. show that the number of submodules of $M \oplus M$ can be infinite.

Answer 1

Consider $M=R=\mathbb{R}$. This is a simple $\mathbb{R}$-module, obviously. Note that for each $\lambda\in\mathbb{R}$ the set $$M_\lambda:=\{(x,\lambda x): x\in\mathbb{R}\}\subset\mathbb{R}\oplus\mathbb{R}$$ is a submodule of $\mathbb{R}\oplus\mathbb{R}$.