Kaplansky's theorem on projective modules

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; 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).

The proofs, both original and later one, use the notion of a countably generated module in an essential way. 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.

Proof

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— Let ${\mathfrak {F}}$ denote the family of modules that are direct sums of some of countably generated submodules (here modules can be those over a ring, a group or even a set of endomorphisms). If $M$ is in ${\mathfrak {F}}$, then each of direct summand of $M$ is also in ${\mathfrak {F}}$.

Proof: Let N be a direct summand; i.e., $M=N\oplus L$. Using the assumption, we write $M=\bigoplus _{I\in I}M_{i}$ where each $M_{i}$ is a countably generated submodule. For each subset $A\subset I$, we write $M_{A}=\bigoplus _{i\in A}M_{i},N_{A}=$ the image of $M_{A}$ under the projection $M\to N\hookrightarrow M$ and $L_{A}$ the same way. Now, consider the set of all triples ($J$, $B$, $C$) consisting of a subset $J\subset I$ and subsets $B,C\subset {\mathfrak {F}}$ such that $M_{J}=N_{J}\oplus L_{J}$ and $N_{J},L_{J}$ are the direct sums of the modules in $B,C$. We give this set a partial ordering such that $(J,B,C)\leq (J',B',C')$ if and only if $J\subset J'$, $B\subset B',C\subset C'$. By Zorn's lemma, the set contains a maximal element $(J,B,C)$. We shall show that $J=I$; i.e., $N=N_{J}=\bigoplus _{N'\in B}N'\in {\mathfrak {F}}$. Suppose otherwise. Then we can inductively construct a sequence of at most countable subsets $I_{1}\subset I_{2}\subset \cdots \subset I$ such that $I_{1}\not \subset J$ and for each integer $n\geq 1$,

$M_{I_{n}}\subset N_{I_{n}}+L_{I_{n}}\subset M_{I_{n+1}}$.

Let $I'=\bigcup _{0}^{\infty }I_{n}$ and $J'=J\cup I'$. We claim:

$M_{J'}=N_{J'}\oplus L_{J'}.$

The inclusion $\subset$ is trivial. Conversely, $N_{J'}$ is the image of $N_{J}+L_{J}+M_{I'}\subset N_{J}+M_{I'}$ and so $N_{J'}\subset M_{J'}$. The same is also true for $L_{J'}$. 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., $N_{J}\oplus M'=M$ for some $M'$. Then, by modular law, $N_{J'}=N_{J}\oplus (M'\cap N_{J'})$. Set ${\widetilde {N_{J}}}=M'\cap N_{J'}$. Define ${\widetilde {L_{J}}}$ in the same way. Then, using the early claim, we have:

$M_{J'}=M_{J}\oplus {\widetilde {N_{J}}}\oplus {\widetilde {L_{J}}},$

which implies that

${\widetilde {N_{J}}}\oplus {\widetilde {L_{J}}}\simeq M_{J'}/M_{J}=M_{J'-J}$

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

Lemma 2—If $M_{i},i\in I$ are countably generated modules with local endomorphism rings and if $N$ is a countably generated module that is a direct summand of $\bigoplus _{i\in I}M_{i}$, then $N$ is isomorphic to $\bigoplus _{i\in I'}M_{i}$ for some subsets $I'\subset I$.

Idea of Proof: Let ${\mathcal {G}}$ denote the family of modules that are of the form $\bigoplus _{i\in F}M_{i}$ for some finite subset $F\subset I$. The assertion is then implied by the following claim:

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

Indeed, assume the claim is valid. Then choose a sequence $x_{1},x_{2},\dots$ in N that is a generating set. Then using the claim, write $N=H_{1}\oplus N_{1}$ where $x_{1}\in H_{1}\in {\mathcal {G}}$. Then we write $x_{2}=y+z$ where $y\in H_{1},z\in N_{1}$. We then decompose $N_{1}=H_{2}\oplus N_{2}$ with $z\in H_{2}\in {\mathcal {G}}$. Note $\{x_{1},x_{2}\}\subset H_{1}\oplus H_{2}$. Repeating this argument, in the end, we have: ${\textstyle \{x_{1},x_{2},\dots \}\subset \bigoplus _{0}^{\infty }H_{n}}$; i.e., ${\textstyle N=\bigoplus _{0}^{\infty }H_{n}}$. Hence, the proof reduces to proving the claim and the claim is a consequence of Azumaya's theorem. $\square$

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 ${\mathfrak {F}}$ 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. $\square$

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.

Theorem— 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 $M=\bigoplus _{i\in I}M_{i}$ such that for each maximal direct summand L of M, there is a decomposition $M={\Big (}\bigoplus _{j\in J}M_{j}{\Big )}\bigoplus L$ for some subset $J\subset I$.

References

• 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
• 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