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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity and let $P$ be a left $R$-[[module|0NZ8JA25WK]].\nThe module $P$ is a *generator* if every left $R$-module is a homomorphic image of a direct sum of copies of $P$.\nThe module $P$ is a *progenerator* if all three of these hold:\n\n1. $P$ is [[finitely generated|05QY1VEBYD]].\n2. $P$ is [[projective|BFFA03GE3V]].\n3. $P$ is a generator.\n\n## Remarks\n\nA module $P$ is a generator if and only if $R$ is a direct summand of $P^n$ for some $n \\ge 1$.\nExample: $P = R^n$ with $n \\ge 1$ is free of finite rank, hence finitely generated and projective, and $R$ is a direct summand of it, so $P$ is a progenerator.\nNon-example: let $k$ be a field, $R = k \\times k$ and $P = k \\times 0 = R(1, 0)$.\nIt is finitely generated and projective, as a direct summand of $R$.\nIt is not a generator: every map $P \\to 0 \\times k$ is zero, since it sends $(1, 0) = (1, 0)^2$ to $(1, 0) \\cdot m = 0$; so $0 \\times k$ is not a quotient of a sum of copies of $P$.\nOver a commutative ring $R$, a finitely generated projective module is a generator if and only if it is faithful, that is, its rank is positive at every prime ideal.\nThe ring $\\End_R(P)$ of a progenerator $P$ is [[Morita equivalent|RD5WKA1VK7]] to $R$.\n",
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