Projectivity of $\mathbb Q$ over $\mathbb Q\otimes_{\mathbb Z}\mathbb Q$

Question

Consider $\mathbb Q\otimes \mathbb Q$, where $\mathbb Q$ is considered as $\mathbb Z$-algebra and consider $\mathbb Q$ as a right $\mathbb Q\otimes\mathbb Q$ module. Then is it true that $\mathbb Q$ is projective $\mathbb Q\otimes\mathbb Q$-module?

Answer 1

Since $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} \cong \mathbb{Q}$ is a field, every module over it is projective.

Answer 2

I assume that the module structure is induced by the ring map $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} \to \mathbb{Q}$, $a \otimes b \mapsto ab$. But this is an isomorphism, so that the module is free of rank $1$. More generally, if $A \to B$ is an epimorphism in the category of commutative rings, then $B \otimes_A B \to B$ is an isomorphism, so that $B$ is free of rank $1$ over $B \otimes_A B$.