## Yet Another Algebra that is not Separable

Let $R$ be a commutative ring and $A$ be an $R$-algebra. We say that $A$ is a separable $R$-algebra if $A$ is projective as an $A\otimes_R A^{\rm op}$-module, where the action of $A\otimes_RA^{\rm op}$ is given by $(a\otimes a')b = aba'$. We already showed that the ring of upper triangular matrices over a commutative ring […]