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  "body": "\n## Summary\n\nA projective cover of a [[module|0NZ8JA25WK]] $M$ is a surjection from a [[projective module|BFFA03GE3V]] onto $M$ that is as small as possible.\nThis means that its kernel is a superfluous submodule.\nWhen it exists it is unique up to isomorphism.\nIt is the dual notion of the [[injective hull|QCJN05GN64]], which always exists, whereas projective covers exist only over special rings.\nThe rings over which every finitely generated module has a projective cover are the [[semiperfect rings|ZRC2S3ECFK]].\nThose over which every left module has one are the left [[perfect rings|49Y9Q949PK]].\n\n## Explanations\n\n### L1 human\n\nEvery module is a quotient of a projective module, for instance of a free module, but usually in many ways and with a lot of waste.\nA projective cover is a most economical way to do it: no proper piece of the projective module already maps onto $M$.\nFor example, over the ring $\\Z/4$, the map $\\Z/4 \\to \\Z/2$ is a projective cover, since the kernel $2\\Z/4\\Z$ is too small to help any proper submodule fill everything.\nOver the integers, the module $\\Z/2$ has no projective cover.\n",
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