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  "body": "\n## Definition\n\nLet $R$ and $S$ be [[rings|0B1261B5ED]] with identity and let $U$ be an $R$-$S$-[[bimodule|EM7FHAFD3N]].\nA left $R$-module $C$ is a *cogenerator* if every left $R$-module embeds in a direct product of copies of $C$; likewise on the right.\nThe bimodule $U$ defines a *Morita duality* between $R$ and $S$ if these hold:\n\n1. $U$ is an [[injective|85H7PRHDPZ]] cogenerator as a left $R$-module and as a right $S$-module.\n2. The map $R \\to \\End(U_S)$ sending $r$ to left multiplication by $r$ is a ring isomorphism.\n3. The map $S \\to \\End({}_R U)^{\\mathrm{op}}$ sending $s$ to right multiplication by $s$ is a ring isomorphism.\n\nHere $\\End(U_S)$ and $\\End({}_R U)$ are the [[endomorphism rings|AWSBZWWMG9]] of $U$ as a right $S$-module and as a left $R$-module.\nA ring $R$ *has a Morita duality* if such $S$ and $U$ exist.\n\n## Remarks\n\nA left $R$-module $M$ is *reflexive* if the evaluation map $M \\to \\Hom_S(\\Hom_R(M, U), U)$ is an isomorphism, and likewise for right $S$-modules.\nUnder a Morita duality, the functors $\\Hom_R(-, U)$ and $\\Hom_S(-, U)$ are mutually inverse contravariant equivalences between the reflexive modules on the two sides.\nThe reflexive modules include all [[finitely generated|05QY1VEBYD]] modules and all finitely cogenerated modules.\nExample: for a field $k$, take $R = S = U = k$.\nThe space $k$ is injective and is a cogenerator of $k$-vector spaces, the multiplication maps $k \\to \\End(k)$ are isomorphisms, and the duality is $V \\mapsto V^*$ on finite-dimensional spaces.\nNon-example: $R = S = U = \\Z$ fails clause 1, since $\\Z$ is not an injective $\\Z$-module; the map $2\\Z \\to \\Z$, $2 \\mapsto 1$, does not extend to $\\Z$.\nOver a quasi-Frobenius ring $R$, the bimodule $U = R$ gives a Morita duality between $R$ and itself.\nOver a complete Noetherian local commutative ring with residue field $k$, the [[injective hull|QCJN05GN64]] of $k$ gives a Morita duality, known as Matlis duality.\n",
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