Let $A$ be an $R$-algebra ($R$ is a commutative ring with identity $1_R$) and suppose $F$ is a left free module over $A$. Is it true that $$M\otimes_A F\simeq M^{(I)}$$ for any right module $M$ over $A$?
Above $\displaystyle M^{(I)}$ means direct sum of copies of $M$.
Obs. If $M$ is a flat module then the above is trivially true. In fact, since $F$ is free $F\simeq A^{(I)}$ and using the exactness of the functor $M\otimes_A-$ one gets $$M\otimes_A F\simeq M\otimes_A A^{(I)}\simeq M^{(I)}.$$
Thanks