Azumaya algebra

Remarks

Example: $A = M_n(R)$ with $n \ge 1$. It is free of rank $n^2$, hence projective and faithful, and the map $M_n(R) \otimes_R M_n(R)^{\mathrm{op}} \to \End_R(M_n(R))$ sends the basis elements $e_{ij} \otimes e_{kl}$ to the maps $x \mapsto e_{ij} x e_{kl}$, which form a basis of $\End_R(M_n(R)) \cong M_{n^2}(R)$. Non-example: let $L/K$ be a field extension of degree $2$ and $A = L$ over $R = K$. Clauses 1 and 2 hold, but $L \otimes_K L$ is commutative while $\End_K(L) \cong M_2(K)$ is not, so the map in clause 3 is not an isomorphism. Over a field $K$, the Azumaya algebras are exactly the central simple algebras over $K$. Equivalently, $A$ is an Azumaya algebra when it is a central separable algebra over $R$. Another equivalent form: clauses 1 and 2 hold and $A / \mathfrak{m}A$ is central simple over $R/\mathfrak{m}$ for every maximal ideal $\mathfrak{m}$ of $R$.