# Module spectrum maps up to stable homotopy

Let $$R$$ be a commutative ring spectrum, $$M$$ and $$N$$ be a $$R$$-module spectra.

Let us consider $$R$$-module maps from $$M$$ to $$N$$ up to stable homotopy, that is maps $$M \to N$$ such that the composites $$R \wedge M \to M \xrightarrow{f} N$$ and $$R \wedge M \xrightarrow{1 \wedge f} R \wedge N \to N$$ are equal in the stable homotopy category.

Now suppose that $$M = R \wedge X$$ is a free $$R$$-module.

Is it true that the set of stable homotopy classes of $$R$$-module maps up to homotopy in the previous sense, $$[M, N]_R = [R \wedge X, N]_R$$, is in the natural bijection with stable homotopy classes of all maps $$[X, N]$$?

I believe that something like that (or even stronger) holds for genuine $$R$$-module maps, but what about homotopy $$R$$-module maps if we are interested in the stable homotopy classes of maps only?