Isomorphic as $R$-modules v.s. Isomorphic as abelian groups

Question

Let $(R,\mathfrak m, \mathbb Q)$ be a Noetherian local ring. Let $M$ be a finitely generated $R$-module such that for some integer $n\ge 0$, there is an isomorphism of abelian groups $M \cong \mathbb Q^{\oplus n}$.

My question is: Is $M$ Isomorphic, as an $R$-module, to a direct sum of copies of $\mathbb Q$? i.e., is it true that $\mathfrak m M=0$?

Answer 1

No : take $R=\mathbb Q[\epsilon] = \mathbb Q[X]/(X^2)$; then $R$ itself satisfies your hypotheses.