A right $R$ module $M$ has finitely many submodules iff right $R$ module $M^n$ has finitely many submodules for any positive integer $n$.

Question

A right $R$ module $M$ has finitely many submodules iff right $R$ module $M^n$ has finitely many submodules for any positive integer $n$.

When $M^n$ has finitely many submodules for any positive integer $n$, then clearly $M$ has finitely many submodules. But, I have failed to prove it's converse. Please help me.

Answer 1

Completely false. Let $R = \mathbb{R}$. Then $\mathbb{R}$ has two submodules, while $\mathbb{R}^2$ has infinitely many [namely, all lines through the origin].