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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:

- 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"
^{[4]}).

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.

**Lemma 1** — ^{[6]} Let 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). If is in , then each direct summand of is also in .

*Proof*: Let *N* be a direct summand; i.e., . Using the assumption, we write where each is a countably generated submodule. For each subset , we write the image of under the projection and the same way. Now, consider the set of all triples ( , , ) consisting of a subset and subsets such that and are the direct sums of the modules in . We give this set a partial ordering such that if and only if , . By Zorn's lemma, the set contains a maximal element . We shall show that ; i.e., . Suppose otherwise. Then we can inductively construct a sequence of at most countable subsets such that and for each integer ,

- .

Let and . We claim:

The inclusion is trivial. Conversely, is the image of and so . The same is also true for . Hence, the claim is valid.

Now, is a direct summand of (since it is a summand of , which is a summand of ); i.e., for some . Then, by modular law, . Set . Define in the same way. Then, using the early claim, we have:

which implies that

is countably generated as . This contradicts the maximality of .

**Lemma 2** — 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 .

*Proof*:^{[7]} Let denote the family of modules that are isomorphic to modules of the form for some finite subset . The assertion is then implied by the following claim:

- Given an element , there exists an that contains
*x*and is a direct summand of*N*.

Indeed, assume the claim is valid. Then choose a sequence in *N* that is a generating set. Then using the claim, write where . Then we write where . We then decompose with . Note . Repeating this argument, in the end, we have: ; i.e., . 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).

*Proof of the theorem*: Let be a projective module over a local ring. Then, by definition, it is a direct summand of some free module . This is in the family in Lemma 1; thus, 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 is countably generated. 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.

**Theorem** — ^{[8]} Let *R* be a ring. Then the following are equivalent.

*R*is a local ring.- Every projective module over
*R*is free and has an indecomposable decomposition such that for each maximal direct summand*L*of*M*, there is a decomposition for some subset .

The implication is exactly (usual) Kaplansky's theorem and Azumaya's theorem. The converse follows from the following general fact, which is interesting itself:

- A ring
*R*is local for each nonzero proper direct summand*M*of , either or .

is by Azumaya's theorem as in the proof of . Conversely, suppose has the above property and that an element *x* in *R* is given. Consider the linear map . Set . Then , which is to say splits and the image is a direct summand of . It follows easily from that the assumption that either *x* or -*y* is a unit element.

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^{a}^{b}Anderson & Fuller 1992, Corollary 26.7. **^**Anderson & Fuller 1992, Proposition 15.15.**^**Matsumura 1989, Theorem 2.5.**^**Lam 2000, Part 1. § 1.**^**Bass 1963**^**Anderson & Fuller 1992, Theorem 26.1.**^**Anderson & Fuller 1992, Proof of Theorem 26.5.**^**Anderson & Fuller 1992, Exercise 26.3.

- Anderson, Frank W.; Fuller, Kent R. (1992),
*Rings and categories of modules*, Graduate Texts in Mathematics, vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3, MR 1245487 - Bass, Hyman (February 28, 1963). "Big projective modules are free".
*Illinois Journal of Mathematics*.**7**(1). University of Illinois at Champagne-Urbana: 24–31. doi:10.1215/ijm/1255637479. - Kaplansky, Irving (1958), "Projective modules",
*Ann. of Math.*, 2,**68**(2): 372–377, doi:10.2307/1970252, hdl:10338.dmlcz/101124, JSTOR 1970252, MR 0100017 - Lam, T.Y. (2000). "Bass's work in ring theory and projective modules". arXiv:math/0002217. MR1732042
- Matsumura, Hideyuki (1989),
*Commutative Ring Theory*, Cambridge Studies in Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6