![]() ![]() Show that a countably generated projective module over a local ring is free (by a " of the proof of Nakayama's lemma" ).Observe that a projective module over an arbitrary ring is a direct sum of countably generated projective modules. ![]() For the general case, the proof (both the original as well as later one) consists of the following two steps: The theorem can also be formulated so to characterize a local ring ( #Characterization of a local ring).įor a finite projective module over a commutative local ring, the theorem is an easy consequence of Nakayama's lemma. ![]() 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. ![]()
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