Kaplansky's theorem on projective modules (original) (raw)
In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; where a not-necessary-commutative ring is called local if for each element x, either x or 1 − x is a unit element. The theorem can also be formulated so to characterize a local ring. For a finite projective module over a commutative local ring, the theorem is an easy consequence of Nakayama's lemma. For the general case, the proof (both the original as well as later one) consists of the following two steps:
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| dbo:abstract | In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; where a not-necessary-commutative ring is called local if for each element x, either x or 1 − x is a unit element. The theorem can also be formulated so to characterize a local ring. For a finite projective module over a commutative local ring, the theorem is an easy consequence of Nakayama's lemma. For the general case, the proof (both the original as well as later one) consists of the following two steps: * Observe that a projective module over an arbitrary ring is a direct sum of countably generated projective modules. * Show that a countably generated projective module over a local ring is free (by a "[reminiscence] of the proof of Nakayama's lemma"). The idea of the proof of the theorem was also later used by Hyman Bass to show (under some mild conditions) are free. According to, Kaplansky's theorem "is very likely the inspiration for a major portion of the results" in the theory of semiperfect rings. (en) |
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| dbp:mathStatement | Let denote the family of modules that are direct sums of some of countably generated submodules . If is in , then each of direct summand of is also in . (en) If are countably generated modules with local endomorphism rings and if is a countably generated module that is a direct summand of , then is isomorphic to for some at most countable subset . (en) |
| dbp:name | Lemma 1 (en) Lemma 2 (en) |
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| rdfs:comment | In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; where a not-necessary-commutative ring is called local if for each element x, either x or 1 − x is a unit element. The theorem can also be formulated so to characterize a local ring. For a finite projective module over a commutative local ring, the theorem is an easy consequence of Nakayama's lemma. For the general case, the proof (both the original as well as later one) consists of the following two steps: (en) |
| rdfs:label | Kaplansky's theorem on projective modules (en) |
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