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  "body": "\n## Definition\n\nLet $R$ and $S$ be [[rings|0B1261B5ED]] with identity.\nLet $R\\text{-Mod}$ be the category of left $R$-[[modules|0NZ8JA25WK]] and $R$-linear maps, and likewise $S\\text{-Mod}$.\nThe rings $R$ and $S$ are *Morita equivalent* if there is an [[equivalence of categories|K7A04ZBD92]] $F \\colon R\\text{-Mod} \\to S\\text{-Mod}$.\n\n## Remarks\n\nExample: for a field $k$, the rings $k$ and $M_2(k)$ are Morita equivalent.\nThe functor sends a vector space $V$ to $V^2 = k^2 \\otimes_k V$, the column vectors with entries in $V$, on which $M_2(k)$ acts by matrix multiplication.\nA quasi-inverse sends a module $N$ to $e_{11} N$, where $e_{11}$ is the matrix unit.\nNon-example: $k$ and $k \\times k$ are not Morita equivalent.\nIn $k\\text{-Mod}$ every nonzero module has a nonzero map to every other nonzero module, while the simple $k \\times k$-modules $k \\times 0$ and $0 \\times k$ have no nonzero map between them.\nAn equivalence of categories preserves this property.\nAn equivalence between module categories is automatically additive and is naturally isomorphic to $P \\otimes_R -$ for an $S$-$R$-bimodule $P$.\nMorita's theorem: $R$ and $S$ are Morita equivalent if and only if $S \\cong \\End_R(P)^{\\mathrm{op}}$ for some [[progenerator|ZQ9ABBKF1N]] $P$ of $R\\text{-Mod}$, where $\\End_R(P)$ acts on $P$ from the left.\nThe definition with right modules gives the same relation.\nIsomorphic rings are Morita equivalent, and for commutative rings the converse holds.\n",
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