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;[1] where a not-necessarily-commutative ring is called local if for each element x, either x or 1 − x is a unit element.
[2] 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 Nakayama's lemma.
[3] For the general case, the proof (both the original as well as later one) consists of the following two steps: The idea of the proof of the theorem was also later used by Hyman Bass to show big projective modules (under some mild conditions) are free.
[5] According to (Anderson & Fuller 1992), Kaplansky's theorem "is very likely the inspiration for a major portion of the results" in the theory of semiperfect rings.
[1] The proof of the theorem is based on two lemmas, both of which concern decompositions of modules and are of independent general interest.
denote the family of modules that are direct sums of some countably generated submodules (here modules can be those over a ring, a group or even a set of endomorphisms).
Proof: Let N be a direct summand; i.e.,
is a countably generated submodule.
are the direct sums of the modules in
We give this set a partial ordering such that
By Zorn's lemma, the set contains a maximal element
Then we can inductively construct a sequence of at most countable subsets
Then, using the early claim, we have: which implies that is countably generated as
are countably generated modules with local endomorphism rings and if
is a countably generated module that is a direct summand of
denote the family of modules that are isomorphic to modules of the form
The assertion is then implied by the following claim: Indeed, assume the claim is valid.
Repeating this argument, in the end, we have:
Hence, the proof reduces to proving the claim and the claim is a straightforward consequence of Azumaya's theorem (see the linked article for the argument).
be a projective module over a local ring.
Then, by definition, it is a direct summand of some free module
is a direct sum of countably generated submodules, each a direct summand of F and thus projective.
Hence, without loss of generality, we can assume
Then Lemma 2 gives the theorem.
Kaplansky's theorem can be stated in such a way to give a characterization of a local ring.
A direct summand is said to be maximal if it has an indecomposable complement.
follows from the following general fact, which is interesting itself:
is by Azumaya's theorem as in the proof of
It follows easily from that the assumption that either x or -y is a unit element.