( Background: These is one part of a critera for commutative rings $ f:A \rightarrow B$ to be etale. )
It is claimed that the following two conditions 1 & 2 are equivalent.
- The multiplication map $$ p: B \otimes _A B \rightarrow B$$ is the projection on to a summand. There exists another commutative ring $q: B \otimes _A B \rightarrow R$ such that $p$ and $q$ induces isomorphism $$ B \otimes_A B \rightarrow B \times R $$
- There exists an idempotent element $e \in B \otimes_A B$ such that $p$ induces an isomorphism $$(B \otimes_A B)[1/e] \simeq B$$ under localization at $e$.
I can prove 1=>2. But I can't prove 2=>1. Help would be appreciated : )