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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity and let $M$ be a left $R$-[[module|0NZ8JA25WK]].\nA [[submodule|05WDSN597C]] $K$ of a module $P$ is *superfluous* if for every submodule $L$ of $P$, $L + K = P$ implies $L = P$.\nA *projective cover* of $M$ is a surjective $R$-linear map $p \\colon P \\to M$ such that:\n\n1. $P$ is a [[projective module|BFFA03GE3V]].\n2. The kernel $\\ker p$ is superfluous in $P$.\n\n## Remarks\n\nIf $p \\colon P \\to M$ and $p' \\colon P' \\to M$ are projective covers, there is an isomorphism $f \\colon P \\to P'$ with $p' \\circ f = p$.\nExample: let $R = \\Z/4$ and $p \\colon R \\to \\Z/2$ the reduction map.\nThe module $R$ is free, hence projective, and $\\ker p = 2R$.\nThe submodules of $R$ are $0$, $2R$ and $R$, and $0 + 2R$ and $2R + 2R$ are not $R$, so $2R$ is superfluous and $p$ is a projective cover.\nNon-example: over $R = \\Z$, the reduction map $\\Z \\to \\Z/2$ has kernel $2\\Z$, and $2\\Z + 3\\Z = \\Z$ with $3\\Z \\ne \\Z$, so the kernel is not superfluous.\nIn fact the $\\Z$-module $\\Z/2$ has no projective cover at all.\nOver a local ring $R$ with maximal ideal $\\mathfrak m$, the map $R \\to R / \\mathfrak m$ is a projective cover, by Nakayama's lemma.\nBass proved two characterizations.\nA ring is semiperfect if and only if every finitely generated left module has a projective cover.\nA ring is left perfect if and only if every left module has a projective cover.\n",
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