Kaplansky's theorem on projective modules

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, Kaplansky's theorem on projective modules, first proven by <a href="/facts/Irving_Kaplansky/nRLyp7K9">Irving Kaplansky</a>, states that a <a href="/facts/Projective_module/lHymqhNC">projective module</a> over a <a href="/facts/Local_ring/gRTdg5Qx">local ring</a> is <a href="/facts/Free_module/o6rC6yoJ">free</a>; where a not-necessarily-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 (#Characterization of a local ring).
For a finite projective module over a commutative local ring, the theorem is an easy consequence of <a href="/facts/Nakayama%2527s_lemma/Nwhck7ru">Nakayama's lemma</a>. For the general case, the proof (both the original as well as later one) consists of the following two steps:

<ul><li>Observe that a projective module over an arbitrary ring is a direct sum of <a href="/facts/Countably_generated_module/b9c4FnWJ">countably generated</a> projective modules.</li>
<li>Show that a countably generated projective module over a local ring is free (by a "[reminiscence] of the proof of Nakayama's lemma").</li></ul>
The idea of the proof of the theorem was also later used by <a href="/facts/Hyman_Bass/Se6dGEd6">Hyman Bass</a> to show big projective modules (under some mild conditions) are free. According to (Anderson & Fuller 1992), Kaplansky's theorem "is very likely the inspiration for a major portion of the results" in the theory of <a href="/facts/Semiperfect_ring/kvcwAEAF">semiperfect rings</a>.

Kaplansky's theorem on projective modules open-in-new

Kaplansky's theorem on projective modules