tensor product of modules over algebra and isomorphism

I have always wondered this statement is true.

Let $$S$$ be commutative algebra over $$R$$ and $$M,N$$ be modules over $$S$$. Then, $$M\otimes_R N\simeq M\otimes_S N$$ as $$S$$-modules. (Or can be as $$R$$-modules.)

Why I think it is true: A map between two modules can be induced by universal property of tensor product; $$m\otimes n\mapsto m\otimes n$$. Surjectivity is trivial and injectivity may be shown by the definition of tensor product. (Actually, it is a quotient module of a free module.)

So what I want to know is: is this known to be true? Or just this is my mistake.

This is false: the only result we have is the canonical homomorphism $$\;M\otimes_R N\longrightarrow M\otimes_SN\;$$ is surjective.
As a counter-example, consider $$\mathbf C$$ as a vector space on $$\mathbf R$$: then $$\dim_\mathbf{R}(\mathbf C\otimes_\mathbf{R}\mathbf C) =4$$, whereas $$\dim_\mathbf{R}(\mathbf C\otimes_\mathbf{C}\mathbf C)=\dim_\mathbf{R}\mathbf C=2$$.