Correlation between Tensor products over a Ring and it's subring

Question

I'm intrested in finding out what the correlation between the two tensor products $$ M \otimes_R N \quad and \quad M \otimes_S N \quad $$ is.(R,S are Rings and S is a subring of R) I know that we can construct a surjective Homomorphism $\phi : M\otimes_R N \to M\otimes_S N$ Now i just used the isomorphism theorem to derive $M\otimes _{ R }N\quad \cong \quad M\otimes _{ S }N/ker\quad \phi) $ Is this really all there is or can we say more things about their correlation