Free Module - sufficient condition

Question

Let $R$ be a commutative ring with unity. Let $I$ be an ideal of $R$. Let $M$ be a left $R$-module. I came across the result that for a free $R$-module $N$, any two bases are equipotent. While proving this result, I stumbled upon the fact that $IN=N$ if and only if $I=R$. I'm interested in knowing whether the converse to this statement holds. That is, if $M$ satisfies the statement $IM=M$ if and only if $I=R$, then is $M$ a free module? If not, can the statement be strengthened to be a sufficient condition for a module to be free? I'm not sure how I should approach it. Any help is much appreciated. Thank you