Abelian Groups - László Fuchs [Springer]

8 Finite Rank p-Local Groups 455
A 7! Hom.G; A/, where Hom is a projective right E-module. Thus Hom.G; A/ n D
Hom.G; C/ n . Passing modulo the Jacobson radical,
ŒHom.G; A/=J Hom.G; A/ n D ŒHom.G; C/=J Hom.G; C/ n :
Since E/J is finite, so artinian, the category of E/J-modules enjoys the Krull-
Schmidt property. Hence we conclude
Hom.G; A/=J Hom.G; A/ Š Hom.G; C/=J Hom.G; C/:
We can argue as in Lemma 6.5 to prove the isomorphy of the projective right
E-modules Hom.G; A/ and Hom.G; C/. Therefore, Hom.G; A/ ˝E G Š A is
isomorphic to Hom.G; C/ ˝E G Š C.
ut
Modules Over the p-Adic Integers The situation becomes extremely simple,
even in the countable rank case, if we move to modules over the ring J p of p-adic
integers.
Theorem 8.4 (Derry [1], Kaplansky [2]). Let M be a torsion-free J p -module of
countable rank. Then M is a direct sum of rank 1 modules, each isomorphic to J p or
to its quotient field Q p .
Proof. Every rank 1 subgroup of M is contained in a pure rank 1 subgroup
isomorphic to J p or Q p . These are algebraically compact, so summands. The
finite rank case is now obvious, while for countable rank the argument used in
Pontryagin’s theorem 7.1 in Chapter 3 establishes the claim.
ut
Let G be an arbitrary torsion-free group, and G .p/ D Z .p/ ˝ G its localization at
the prime p. It is harmless to identify G .p/ with the subgroup of the divisible hull
D of G whose elements x satisfy nx 2 G for some n 2 N coprime to p. Wethen
have G D\ p G .p/ where the intersection is taken in the injective hull of G (see
Lemma 5.1 of Chapter 8).
If we embed the divisible hull D of G (which is a Q-vector space) in the Q p -
vector space Q p ˝ D D Q p ˝ G via d 7! 1 ˝ d, then we get an embedding of G
in J p ˝ G which we will denote by G .p/ .
Lemma 8.5. For a torsion-free group G and a prime p,
G .p/ D D \ G .p/ ;
where the intersection is computed in Q p ˝ G.
Proof. To prove the non-trivial part of the claim that D \ G .p/ G .p/, letx belong
to the intersection. If S Dfg i g i2I is a maximal independent set in G .p/ , then we can