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  "body": "\n## Summary\n\nA progenerator over a [[ring|0B1261B5ED]] $R$ is a [[finitely generated|05QY1VEBYD]] [[projective module|BFFA03GE3V]] that is also a generator of the module category.\nThe free modules $R^n$ with $n \\ge 1$ are progenerators.\nProgenerators are the modules that control [[Morita equivalence|RD5WKA1VK7]].\nA ring $S$ is Morita equivalent to $R$ exactly when it is the endomorphism ring of a progenerator over $R$.\nThis holds up to the choice of side.\n\n## Explanations\n\n### L1 human\n\nThe ring $R$ itself, seen as a module, has three key features.\nIt is generated by one element, it is projective, and every module is a quotient of a sum of copies of it.\nA progenerator is any module that shares these features, with finitely many generators allowed.\nFor example, $R^3$ is a progenerator, since $R$ is a direct summand of it.\nThe ring of endomorphisms of a progenerator has the same module theory as $R$.\n",
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