Abelian Groups - László Fuchs [Springer]

40 1 Fundamentals
For two R-modules, M and N,anR-homomorphism is a group homomorphism
W M ! N that respects multiplication by elements r 2 R, i.e..ra/ D r.a/ for
all r 2 R and a 2 M. The meaning of R-isomorphism, etc. should be obvious.
Example 8.1.
(a) If R is a field, then an R-module is just an R-vector space. In abelian group theory, vector
spaces over the prime fields (Q and Z=pZ) are ubiquitous.
(b) If R is the ring Z of integers, then every abelian group A can be viewed as a Z-module under
the natural definition of multiplication of a 2 A by n 2 Z: na is the nth multiple of a.
Occasionally, we will deal with p-local groups. These are exactly those groups
in which the elements are uniquely divisible by every prime ¤ p. They are genuine
Z .p/ -modules, i.e., modules over the localization of Z at the prime p. The torsion
subgroup t.A/ of a p-local group A is a p-group and A=t.A/ is q-divisible torsionfree
for all primes q ¤ p.
Example 8.2.
(a) The group J p is p-local, and so is the additive group of Z .p/ .
(b) For any group A, the tensor product A ˝ Z .p/ is a p-local group.
p-adic Modules Let us say a few words about modules over the rings Z .p/
and J p . The following observation will be used frequently.
Lemma 8.3. Every p-group is, in a natural way, a module over the ring J p of the
p-adic integers.
Proof. If D s 0 Cs 1 pCs 2 p 2 CCs n p n C2J p ,andifa 2 A is of order p n ,then
a D .s 0 Cs 1 pCs 2 p 2 CCs n 1 p n 1 /a is the natural definition. The element on the
right does not change if we use a longer partial sum for . The module properties
are pretty clear.
Modules over J p are called p-adic modules. For any group A, J p ˝ A is a p-adic
module, and a 7! 1 ˝ a .a 2 A/ is the canonical map W A ! J p ˝ A. is
universal for A in the sense that if M is any p-adic module and ˛ W A ! M any
homomorphism, then there is a unique J p -map W J p ˝ A ! M such that ˛ D .
F Notes. Though several theorems in abelian group theory can be phrased more naturally as
statements on modules over integral domains, or just over Z .p/ or J p , we hesitate to enter unexplored
territory, and will phrase the results to abelian groups only. In this way, inevitably some flavor
is lost, but strict limitations had to be honored. The only exceptions will be cases when p-adic
modules will emerge naturally.
Exercises
(1) For any ring R, the cyclic left R-module Ra is R-isomorphic left annihilator
Ann a of a.
(2) For a submodule N of an R-module M, Anna Ann.a C N/ for all a 2 M.
(3) If R is an integral domain, then the elements a of an R-module M with
Ann a ¤ 0 form a submodule in M (called the torsion submodule).
(4) Every torsion group is a module over the completion QZ of Z in the Z-adic
topology. This ring is the cartesian product of the rings J p for all primes p.