## Example: Separability Idempotent for a Field Extension

Let $R$ be a commutative ring and let $A$ be an $R$-algebra. We say that $A$ is separable if $A$ is projective as an $A\otimes_RA^{\rm op}$-module. There is a multiplication map $\mu:A\otimes_RA^{\rm op}\to A$ given by $a\otimes a'\mapsto aa'$, whose kernel we'll call $J$. It's a fact that $A$ is separable if and only if […]