let be $R$ a commutative ring

Auslander and Goldman define a central separabl $R$-algebra $A$ as a faithful $R$-module with $Z(A) = R$ , and such that $A$ is a projective $A\otimes_R A^{\mathrm{op}}$-module

Following Milne; an Azumaya algebra $A$ over a local ring $R $ is an algebra (with $Z(A) = R$) which is a free module of finite rank over $R$ such that the natural map $A\otimes_R A^{\mathrm{op}}\to \mathrm{End}_R(A)$ is an isomorphism

How to prove that a central separable algebra over a local ring is just an Azumaya algebra.