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  "body": "\n## Statement\n\n1. Let $R$ be a [[ring|0894J980RK]] with identity. Every [[projective|CRE6VEKT32]] left $R$-module is a direct sum of countably generated projective left $R$-modules.\n2. Let $R$ be a local ring, that is, a ring with identity whose quotient $R/J(R)$ by the [[Jacobson radical|XA0DCNH7WY]] is a [[division ring|0WZXK05ME7]]. Every projective left $R$-module is [[free|02E1T12PP4]].\n\n## Remarks\n\nPart 1 is proved by a transfinite construction that builds a projective module up from countably generated direct summands.\nFor part 2 one reduces to countably generated modules by part 1.\nA finitely generated projective module over a local ring is free by Nakayama's lemma.\nIn the countably generated case, an inductive argument splits off free summands one element at a time.\nThe point of part 2 is that no finiteness hypothesis is needed.\nExample: every projective module over the [[local ring|MZ02BKPHJA]] $\\Z_{(p)}$ is free.\nThe local hypothesis is necessary.\nOver $\\Z/6\\Z \\cong \\Z/2\\Z \\times \\Z/3\\Z$, the module $\\Z/2\\Z$ is projective, since it is a direct summand of $\\Z/6\\Z$.\nIt is not free, since it has $2$ elements while a nonzero free module has at least $6$.\n\n## Explanations\n\n### L1 human\n\nProjective modules are pieces of free modules.\nOver a local ring, a ring with a single \"point\", such pieces are always free themselves, even when they are very large.\nThe first part says that every projective module can be cut into pieces that are each generated by countably many elements.\n\n### L2 human\n\nFor finitely generated modules, part 2 is a standard consequence of Nakayama's lemma; Kaplansky's contribution is the infinitely generated case.\nPart 1 is a general reduction used in many other questions about large [[projective modules|BFFA03GE3V]].\nOver [[semiperfect rings|ZRC2S3ECFK]] a related result holds: every finitely generated projective module is a direct sum of modules $Re$ with $e$ an idempotent such that $eRe$ is local.\n",
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