Let $R$ be a commutative ring and $S\subset R$ a subring. Let $M$ and $N$ be two $R$-modules.
1) Via restriction of scalars, we can also view $M$ and $N$ as $S$-modules. Show that we have a surjective homomorphism $\phi:M\otimes_S N\to M\otimes_R N,m\otimes n\mapsto m\otimes n$. Describe its kernel.
2) Let $R=\Bbb C[x]/(x^3)$. Consider $M=\Bbb C\oplus \Bbb C$ as $R$-module by $Xe_1=e_2$ and $Xe_2=0$. Prove that $$M\otimes_R M\cong M.$$
For (1), by definition of the tensor product, the map $M\times N\to M\otimes_R N,(m,n)\mapsto m\otimes n$ is $R$-bilinear. As $S\subset R$ is a subring, it is also $S$-bilinear, so the universal property gives a homomorphism $M\otimes_S N\to M\otimes_R N,m\otimes n\mapsto m\otimes n$. The fact that it is surjective seems trivial to me by the formula, but I feel like I am wrong. For the kernel, I have no idea.
(2) Should this be a consequence of (1)? I tried to use the universal property.
Could anybody help?