There are several equivalent definitions of Azumaya algebras over commutative rings, but they always seem to include a finiteness/projectiveness assumption.
For instance, an Azumaya algebra over $R$ is a $R$-algebra $A$ that is a finite projective $R$-module such that the natural "sandwich" map $$A\otimes_R A^{op}\to \operatorname{End}_R(A)$$ is an isomorphism.
I totally understand why we would want Azumaya algebras to be projective of finite type, but is it necessary to explicitely assume it ?
Are there examples of non-projective algebras with the above property ?