In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, Kaplansky's theorem on projective modules, first proven by
Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andr ...
, states that a
projective module over a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
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. 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. 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 "
eminiscenceof the proof of Nakayama's lemma").
The idea of the proof of the theorem was also later used by
Hyman Bass
Hyman Bass (; born October 5, 1932). The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the c ...
References
External links
*Directory page at University of MichiganAuthor profilein the database zbMATH
{{DEFAUL ...
to show
big projective modules (under some mild conditions) are free. According to , Kaplansky's theorem "is very likely the inspiration for a major portion of the results" in the theory of
semiperfect rings.
Proof
The proof of the theorem is based on two lemmas, both of which concern decompositions of modules and are of independent general interest.
''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
.
''Proof'':
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.
Characterization of a local ring
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.
The implication
is exactly (usual) Kaplansky's theorem and Azumaya's theorem. The converse
follows from the following general fact, which is interested 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.
See also
*
Krull–Schmidt category In category theory, a branch of mathematics, a Krull–Schmidt category is a generalization of categories in which the Krull–Schmidt theorem holds. They arise, for example, in the study of finite-dimensional module (mathematics), modules over an a ...
Notes
References
*
*H. Bass: Big projective modules are free, Illinois J. Math. 7(1963), 24-31.
*
* Y. Lam, Bass’s work in ring theory and projective modules
R 1732042* {{Citation , last1=Matsumura , first1=Hideyuki , title=Commutative Ring Theory , publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambridge University Pre ...
, edition=2nd , series=Cambridge Studies in Advanced Mathematics , isbn=978-0-521-36764-6 , year=1989
Theorems in ring theory
Module theory