---
tag: BG86NXRJ5F
kind: statement
role: theorem
name: Kaplansky's theorem on projective modules
msc: [16D40, 16L30]
uses: [0894J980RK, CRE6VEKT32, 02E1T12PP4, XA0DCNH7WY, 0WZXK05ME7]
about: [BFFA03GE3V]
sources:
  - url: https://en.wikipedia.org/wiki/Kaplansky%27s_theorem_on_projective_modules
same_as:
  wikidata: Q85773557
  wikipedia: Kaplansky's theorem on projective modules
---

## Statement

1. Let $R$ be a [[ring|0894J980RK]] with identity. Every [[projective|CRE6VEKT32]] left $R$-module is a direct sum of countably generated projective left $R$-modules.
2. Let $R$ be a local ring, that is, a ring with identity whose quotient $R/J(R)$ by the [[Jacobson radical|XA0DCNH7WY]] is a [[division ring|0WZXK05ME7]]. Every projective left $R$-module is [[free|02E1T12PP4]].

## Remarks

Part 1 is proved by a transfinite construction that builds a projective module up from countably generated direct summands.
For part 2 one reduces to countably generated modules by part 1.
A finitely generated projective module over a local ring is free by Nakayama's lemma.
In the countably generated case, an inductive argument splits off free summands one element at a time.
The point of part 2 is that no finiteness hypothesis is needed.
Example: every projective module over the [[local ring|MZ02BKPHJA]] $\Z_{(p)}$ is free.
The local hypothesis is necessary.
Over $\Z/6\Z \cong \Z/2\Z \times \Z/3\Z$, the module $\Z/2\Z$ is projective, since it is a direct summand of $\Z/6\Z$.
It is not free, since it has $2$ elements while a nonzero free module has at least $6$.

## Explanations

### L1 human

Projective modules are pieces of free modules.
Over a local ring, a ring with a single "point", such pieces are always free themselves, even when they are very large.
The first part says that every projective module can be cut into pieces that are each generated by countably many elements.

### L2 human

For finitely generated modules, part 2 is a standard consequence of Nakayama's lemma; Kaplansky's contribution is the infinitely generated case.
Part 1 is a general reduction used in many other questions about large [[projective modules|BFFA03GE3V]].
Over [[semiperfect rings|ZRC2S3ECFK]] a related result holds: every finitely generated projective module is a direct sum of modules $Re$ with $e$ an idempotent such that $eRe$ is local.