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  "body": "\n## Summary\n\nA Morita duality between two [[rings|0B1261B5ED]] $R$ and $S$ is a contravariant equivalence between suitable categories of left $R$-modules and right $S$-modules.\nIt is given by a bimodule $U$ through the functors $\\Hom(-, U)$.\nIt generalizes the duality $V \\mapsto V^*$ between finite-dimensional vector spaces.\nMatlis duality over complete Noetherian local rings and the duality $\\Hom(-, R)$ over quasi-Frobenius rings are the main examples.\nMorita characterized such dualities by the bimodule $U$, which must be an [[injective|85H7PRHDPZ]] cogenerator on both sides.\n\n## Explanations\n\n### L1 human\n\nFor finite-dimensional vector spaces, taking the dual space reverses all arrows and taking it twice gives back the space.\nA Morita duality is the same phenomenon for modules over a ring.\nOne fixes a module $U$ that plays the role of the field and sends a module $M$ to the module of maps from $M$ to $U$.\nFor finitely generated modules, doing this twice gives back $M$.\nNot every ring admits such a duality; it forces strong finiteness conditions on the ring.\n",
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