Abelian Groups - László Fuchs [Springer]

3 Endomorphism Rings of Torsion-Free Groups 631
(e) Moving to the global case, suppose R is as stated in the theorem. We get
QR D Q Q p
R .p/ where R Q .p/ is the p-adic completion of the reduced part of the
localization R .p/ D Z .p/ ˝ R. Fora 2 R we write a D .:::;a p ;:::/ with
a p 2 R .p/ .
Just as in (b), for each a 2 R choose a ; a now as a D .:::; pa ;:::/; a D
.:::; pa ;:::/with pa ; pa 2 J p algebraically independent over S .p/ for each p;
note that if R .p/ D 0,then pa D pa D 0 can be chosen. Defining e a as in (16.3)
and A as in (16.4), A becomes a countable subgroup of R. Q As in (c), we argue
that R is isomorphic to a subring of End A. Every 2 End A extends uniquely
to Q 2 End. R Q C / which must act coordinate-wise in each R Q .p/ , because these are
fully invariant subrings in R. Q By the local case, Q is left multiplication by the
R .p/ -component of 1 D c 2 R, thus Q must agree with the left multiplication
by c on all of R, Q in particular on A. This establishes the claim that End A Š R.
ut
Since there is a set of cardinality 2 @ 0
of pairwise disjoint finite subsets of
algebraically independent elements of J p , and since these define non-isomorphic
torsion-free groups in the above construction, it is clear that there are 2 @ 0
nonisomorphic
solutions in Theorem 3.3.
The Topological Version Another point of interest emerges if the endomorphism
rings are equipped with the finite topology. Then all endomorphism rings
of countable reduced torsion-free groups can be characterized, even if they are
uncountable. Note that the necessity of the condition stated in the next theorem
is immediate: each left ideal L in Theorem 3.4 is defined to consist of all 2 End A
that annihilate a fixed a 2 A. However, the proof of sufficiency involves more ring
theory than we care to get into, and therefore we state the theorem without proof.
Theorem 3.4 (Corner [4]). A topological ring R is isomorphic to End Afora
countable reduced torsion-free group A if and only if it is complete in the topology
with a base of neighborhoods of 0 consisting of left ideals L n .n