Kaplansky's theorem on projective modules

The proof of the theorem is based on two lemmas, both of which concern decompositions of modules and are of independent general interest.

{{math_theorem|name=Lemma 1|math_statement={{harvnb|Anderson|Fuller|1992|loc=Theorem 26.1.}} Let $\backslash mathfrak\{F\}$ 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 $M$ is in $\backslash mathfrak\{F\}$, then each direct summand of $M$ is also in $\backslash mathfrak\{F\}$.}}

Proof: Let N be a direct summand; i.e., $M\; =\; N\; \backslash oplus\; L$. Using the assumption, we write $M\; =\; \backslash bigoplus\_\{i\; \backslash in\; I\}\; M\_i$ where each $M\_i$ is a countably generated submodule. For each subset $A\; \backslash subset\; I$, we write $M\_A\; =\; \backslash bigoplus\_\{i\; \backslash in\; A\}\; M\_i,\; N\_A\; =$ the image of $M\_A$ under the projection $M\; \backslash to\; N\; \backslash hookrightarrow\; M$ and $L\_A$ the same way. Now, consider the set of all triples ($J$, $B$, $C$) consisting of a subset $J\; \backslash subset\; I$ and subsets $B,\; C\; \backslash subset\; \backslash mathfrak\{F\}$ such that $M\_J\; =\; N\_J\; \backslash oplus\; L\_J$ and $N\_J,\; L\_J$ are the direct sums of the modules in $B,\; C$. We give this set a partial ordering such that $(J,\; B,\; C)\; \backslash le\; (J\text{'},\; B\text{'},\; C\text{'})$ if and only if $J\; \backslash subset\; J\text{'}$, $B\; \backslash subset\; B\text{'},\; C\; \backslash subset\; C\text{'}$. By Zorn's lemma, the set contains a maximal element $(J,\; B,\; C)$. We shall show that $J\; =\; I$; i.e., $N\; =\; N\_J\; =\; \backslash bigoplus\_\{N\text{'}\; \backslash in\; B\}\; N\text{'}\; \backslash in\; \backslash mathfrak\{F\}$. Suppose otherwise. Then we can inductively construct a sequence of at most countable subsets $I\_1\; \backslash subset\; I\_2\; \backslash subset\; \backslash cdots\; \backslash subset\; I$ such that $I\_1\; \backslash not\backslash subset\; J$ and for each integer $n\; \backslash ge\; 1$,

:$M\_\{I\_n\}\; \backslash subset\; N\_\{I\_n\}\; +\; L\_\{I\_n\}\; \backslash subset\; M\_\{I\_\{n+1\}\}$.

Let $I\text{'}\; =\; \backslash bigcup\_0^\backslash infty\; I\_n$ and $J\text{'}\; =\; J\; \backslash cup\; I\text{'}$. We claim:

:$M\_\{J\text{'}\}\; =\; N\_\{J\text{'}\}\; \backslash oplus\; L\_\{J\text{'}\}.$

The inclusion $\backslash subset$ is trivial. Conversely, $N\_\{J\text{'}\}$ is the image of $N\_J\; +\; L\_J\; +\; M\_\{I\text{'}\}\; \backslash subset\; N\_J\; +\; M\_\{I\text{'}\}$ and so $N\_\{J\text{'}\}\; \backslash subset\; M\_\{J\text{'}\}$. The same is also true for $L\_\{J\text{'}\}$. Hence, the claim is valid.

Now, $N\_J$ is a direct summand of $M$ (since it is a summand of $M\_J$, which is a summand of $M$); i.e., $N\_J\; \backslash oplus\; M\text{'}\; =\; M$ for some $M\text{'}$. Then, by modular law, $N\_\{J\text{'}\}\; =\; N\_J\; \backslash oplus\; (M\text{'}\; \backslash cap\; N\_\{J\text{'}\})$. Set $\backslash widetilde\{N\_J\}\; =\; M\text{'}\; \backslash cap\; N\_\{J\text{'}\}$. Define $\backslash widetilde\{L\_J\}$ in the same way. Then, using the early claim, we have:

:$M\_\{J\text{'}\}\; =\; M\_J\; \backslash oplus\; \backslash widetilde\{N\_J\}\; \backslash oplus\; \backslash widetilde\{L\_J\},$

which implies that

:$\backslash widetilde\{N\_J\}\; \backslash oplus\; \backslash widetilde\{L\_J\}\; \backslash simeq\; M\_\{J\text{'}\}\; /\; M\_J\; \backslash simeq\; M\_\{J\text{'}\; -\; J\}$

is countably generated as $J\text{'}\; -\; J\; \backslash subset\; I\text{'}$. This contradicts the maximality of $(J,\; B,\; C)$. $\backslash square$

{{math_theorem|name=Lemma 2|math_statement=If $M\_i,\; i\; \backslash in\; I$ are countably generated modules with local endomorphism rings and if $N$ is a countably generated module that is a direct summand of $\backslash bigoplus\_\{i\; \backslash in\; I\}\; M\_i$, then $N$ is isomorphic to $\backslash bigoplus\_\{i\; \backslash in\; I\text{'}\}\; M\_i$ for some at most countable subset $I\text{'}\; \backslash subset\; I$.}}

Proof:{{harvnb|Anderson|Fuller|1992|loc=Proof of Theorem 26.5.}} Let $\backslash mathcal\{G\}$ denote the family of modules that are isomorphic to modules of the form $\backslash bigoplus\_\{i\; \backslash in\; F\}\; M\_i$ for some finite subset $F\; \backslash subset\; I$. The assertion is then implied by the following claim:

• Given an element $x\; \backslash in\; N$, there exists an $H\; \backslash in\; \backslash mathcal\{G\}$ that contains x and is a direct summand of N.

Indeed, assume the claim is valid. Then choose a sequence $x\_1,\; x\_2,\; \backslash dots$ in N that is a generating set. Then using the claim, write $N\; =\; H\_1\; \backslash oplus\; N\_1$ where $x\_1\; \backslash in\; H\_1\; \backslash in\; \backslash mathcal\{G\}$. Then we write $x\_2\; =\; y\; +\; z$ where $y\; \backslash in\; H\_1,\; z\; \backslash in\; N\_1$. We then decompose $N\_1\; =\; H\_2\; \backslash oplus\; N\_2$ with $z\; \backslash in\; H\_2\; \backslash in\; \backslash mathcal\{G\}$. Note $\backslash \{\; x\_1,\; x\_2\; \backslash \}\; \backslash subset\; H\_1\; \backslash oplus\; H\_2$. Repeating this argument, in the end, we have: $\backslash \{\; x\_1,\; x\_2,\; \backslash dots\; \backslash \}\; \backslash subset\; \backslash bigoplus\_0^\backslash infty\; H\_n$; i.e., $N\; =\; \backslash bigoplus\_0^\backslash infty\; H\_n$. 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). $\backslash square$

Proof of the theorem: Let $N$ be a projective module over a local ring. Then, by definition, it is a direct summand of some free module $F$. This $F$ is in the family $\backslash mathfrak\{F\}$ in Lemma 1; thus, $N$ 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 $N$ is countably generated. Then Lemma 2 gives the theorem. $\backslash square$

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.

{{math_theorem|math_statement={{harvnb|Anderson|Fuller|1992|loc=Exercise 26.3.}} Let R be a ring. Then the following are equivalent.

1. R is a local ring.
2. Every projective module over R is free and has an indecomposable decomposition $M\; =\; \backslash bigoplus\_\{i\; \backslash in\; I\}\; M\_i$ such that for each maximal direct summand L of M, there is a decomposition $M\; =\; \backslash Big(\backslash bigoplus\_\{j\; \backslash in\; J\}\; M\_j\backslash Big)\; \backslash bigoplus\; L$ for some subset $J\; \backslash subset\; I$.}}

The implication $1.\; \backslash Rightarrow\; 2.$ is exactly (usual) Kaplansky's theorem and Azumaya's theorem. The converse $2.\; \backslash Rightarrow\; 1.$ follows from the following general fact, which is interesting itself:

• A ring R is local $\backslash Leftrightarrow$ for each nonzero proper direct summand M of $R^2\; =\; R\; \backslash times\; R$, either $R^2\; =\; (0\; \backslash times\; R)\; \backslash oplus\; M$ or $R^2\; =\; (R\; \backslash times\; 0)\; \backslash oplus\; M$.

$(\backslash Rightarrow)$ is by Azumaya's theorem as in the proof of $1.\; \backslash Rightarrow\; 2.$. Conversely, suppose $R^2$ has the above property and that an element x in R is given. Consider the linear map $\backslash sigma:R^2\; \backslash to\; R,\; \backslash ,\; \backslash sigma(a,\; b)\; =\; a\; -\; b$. Set $y\; =\; x\; -\; 1$. Then $\backslash sigma(x,\; y)\; =\; 1$, which is to say $\backslash eta:\; R\; \backslash to\; R^2,\; a\; \backslash mapsto\; (ax,\; ay)$ splits and the image $M$ is a direct summand of $R^2$. It follows easily from that the assumption that either x or -y is a unit element. $\backslash square$

• Krull–Schmidt category

{{reflist}}

• {{citation |last1=Anderson|first1=Frank W. |last2=Fuller|first2=Kent R. |title=Rings and categories of modules |series=Graduate Texts in Mathematics |volume=13 |edition=2 |publisher=Springer-Verlag |place=New York |year=1992 |pages=x+376 |isbn=0-387-97845-3 |mr=1245487 |doi=10.1007/978-1-4612-4418-9}}
• {{cite journal |first=Hyman |last=Bass |title= Big projective modules are free |journal=Illinois Journal of Mathematics |publisher=University of Illinois at Champagne-Urbana |date=February 28, 1963 |volume=7 |issue=1 |pages=24–31 |doi=10.1215/ijm/1255637479 |url=https://typeset.io/papers/big-projective-modules-are-free-54r4kq564h|doi-access=free }}
• {{citation|last=Kaplansky|first=Irving|title=Projective modules|journal=Ann. of Math. |series= 2|volume=68|issue=2|year=1958|pages=372–377|mr=0100017|doi=10.2307/1970252|jstor=1970252|hdl=10338.dmlcz/101124|hdl-access=free}}
• {{cite arXiv |eprint=math/0002217 |first=T.Y. |last=Lam |title=Bass's work in ring theory and projective modules |date=2000}} {{MR|1732042}}
• {{Citation | last1=Matsumura | first1=Hideyuki | title=Commutative Ring Theory | publisher=Cambridge University Press | edition=2nd | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-36764-6 | year=1989}}

Category:Theorems in ring theory

Category:Module theory