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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $R$ with identity.\nLet $A^{\\mathrm{op}}$ be the [[opposite ring|M8Q24XB0AF]] of $A$.\nThe algebra $A$ is an *Azumaya algebra* over $R$ if the following hold.\n\n1. $A$ is a [[finitely generated|0XHM30WBVB]] [[projective|CRE6VEKT32]] $R$-module.\n2. $A$ is a [[faithful|TB40Y3SNSX]] $R$-module.\n3. The $R$-linear map $A \\otimes_R A^{\\mathrm{op}} \\to \\End_R(A)$ that sends $a \\otimes b$ to the map $x \\mapsto axb$ is an isomorphism.\n\nHere $\\otimes_R$ is the [[tensor product of modules|SMFZN9E2ZB]] and $\\End_R(A)$ is the [[endomorphism ring|AWSBZWWMG9]] of the $R$-module $A$.\n\n## Remarks\n\nExample: $A = M_n(R)$ with $n \\ge 1$.\nIt 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 $\\End_R(M_n(R)) \\cong M_{n^2}(R)$.\nNon-example: let $L/K$ be a field extension of degree $2$ and $A = L$ over $R = K$.\nClauses 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.\nOver a field $K$, the Azumaya algebras are exactly the [[central simple algebras|5HRKJBAY2V]] over $K$.\nEquivalently, $A$ is an Azumaya algebra when it is a central [[separable algebra|X8H7A98Q55]] over $R$.\nAnother 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 $R$.\n",
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