Abelian Groups - László Fuchs [Springer]

448 12 Torsion-Free Groups
Proof. Let fx 1 ;:::;x n g be a maximal independent set in the additive group of R.
There are t ijk 2 Q such that x i x j D t ij1 x 1 CCt ijn x n .Ifm 2 Z is a common
denominator of the t ijk , then the subgroup generated by the elements y i D mx i is a
subring S of R with free additive group. If R has identity, it can be adjoined to S. ut
The Jordan–Zassenhaus Lemma So far we have managed quite well with the
proofs, but now we have come to a deep result about the finiteness of certain right
modules, without which it seems we cannot prove Lady’s theorem. This result is
fundamental in representation theory (we state it only for the ring Z rather than for
Dedekind domains).
Theorem 6.7 (Jordan–Zassenhaus Lemma (for Z)). Let R be a finitedimensional
semisimple Q-algebra, and S a subring of R such that the additive
groups of S and R have the same rank. For any right R-module H, there are but
finitely many non-isomorphic right S-submodules M of H such that MR D H. ut
This result fits perfectly to verify the following lemma.
Lemma 6.8. A torsion-free ring R of finite rank has but a finite number of nonisomorphic
left ideals that are summands of R.
Proof. In view of Lemma 6.5, the proof can be reduced to the case when R has
0 radical. In this case QR is a semi-simple artinian Q-algebra. Let S be a subring
of QR as stated in Theorem 6.7. Evidently, S can be replaced by nS R with 1
adjoined, i.e. S R can be assumed. As a semi-simple artinian ring, QR has up to
isomorphism but a finite number of right ideals. By Theorem 6.7, there exist only a
finite number of non-isomorphic S-modules in QR, say,M 1 ;:::;M t .
If the right ideal L is a summand of R, thenL D R for an idempotent . For
some integer n >0,wehaven 2 S; and there is an S-isomorphism nS ! M i
for some i. This extends to an R-isomorphism nR ! M i R.AsnR Š R as right
R-modules, the proof is complete.
ut
Lady’s Theorem The lemma just proved, if combined with Lemma 6.1, leads
us at once to the following major result.
Theorem 6.9 (Lady [3]). A torsion-free group of finite rank can have but a finite
number of non-isomorphic direct summands.
ut
It seem sensible at this point to illustrate how the situation is drastically simplified
if we move to J p -modules. There exist only two non-isomorphic rank 1 torsion-free
J p -modules, viz. J p and its field of quotients, Q p . Both are algebraically compact,
and as a consequence, every finite rank torsion-free J p -module is a direct sum of a
finite number of copies of these two modules, so completely decomposable. (By the
way, this also holds if the rank is countable.) Moreover, such direct decompositions
are unique up to isomorphism—this is rather obvious in this situation, and also
follows from the fact that these modules have the exchange property. We can claim
that the torsion-free J p -modules of finite rank have the Krull–Schmidt property.
F Notes. The Lady’s theorem is one of the most important results in the theory of torsion-free
groups. A group-theoretical proof would be most welcome.