All rings I'll consider will be commutative with identity.
Given a homomorphism $f:R \to S$ we can give an $S$-module an $R$-module structure via restriction of scalars. In particular, $S$ can be thought of as an $R$-module with action $$r \circ s = f(r) \cdot s$$
I've long thought that, as an $R$-module, $S \otimes_R S \simeq S$, since it is the quotient of $S \otimes_S S$ by the ideal genreated by relations $(s_1 \circ r) \otimes s_2 = s_1 \otimes (r \circ s_2)$, or equivalently $f(r) \cdot s_1 \otimes s_2 = s_1 \otimes f(r) \cdot s_2$. Since $S \otimes_S S \simeq S$ via the map $s_1 \otimes s_2 \mapsto s_1 s_2$ it seemed like the ideal we were quotienting by was trivial.
Thinking about this though, $\mathbb{C} \otimes_R \mathbb{C} \simeq \mathbb{C} \oplus \mathbb{C}$
Are there any conditions on $R,S$ that would make $S \otimes_R S \simeq S$, as an $R$-module? For example $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} \simeq \mathbb{Q}$ holds, as well as $\mathbb{Z}/p \otimes_{\mathbb{Z}} \mathbb{Z}/p \simeq \mathbb{Z}/p$
Best Answer
The generalizations of the two examples you gave are:
I'm sure other conditions are possible. I don't know of general conditions which classify when this does or does not hold.