Abelian Groups - László Fuchs [Springer]

674 18 Groups in Rings and in Fields
additive group is isomorphic to the group A will be called a ring on A; wealso
say R is supported by A.
Lemma 1.1. For all a; b 2 R and m; n 2 N, we have:
(a) If ma D 0 and nb D 0, then dab D 0 where d D gcdfm; ng.
(b) If mja and mb D 0,thenabD 0.
(c) mja and njb imply mnjab.
Proof. Follow the proof of Lemma 1.2 in Chapter 8.
ut
Some immediate corollaries of this lemma are listed below.
(A) In every ring R, the following are (two-sided) ideals:
(i) nR and RŒn for every n 2 N;
(ii) the torsion part t.R/, and its p-components;
(iii) the Ulm subgroups, the subgroup p R for every ordinal , including the
p-divisible part p 1 R and the divisible subgroup of R;
(iv) the fully invariant subgroups of the additive group R C ;
(v) if R is torsion-free, then R(t)isanidealforeverytypet.
Similar statements hold if R is replaced by a left or right ideal. (iv) follows
from the fact that in any ring R, multiplication by an element from the left (or
from the right) is an endomorphism of R C .
(B) If R is a torsion ring, then there is a ring direct decomposition R D˚p R p into
p-components.
(C) If pje for an idempotent e, thenp k je for all k 2 N. If the ring R has an identity
1,thenmj1 .m 2 N/ implies m k ja for all a 2 R and k 2 N:
(D) From (C) we obtain that the p-heights satisfy
h p .ab/ h p .a/ C h p .b/ for all a; b 2 R:
(E) Consequently, if R is a torsion-free ring, then the characters obey the rule
.ab/ .a/.b/ for all a; b 2R.
(F) Elements in the torsion subgroup of a ring are annihilated by elements in the
first Ulm subgroup, in particular, by the elements in the divisible part of the
additive group. The additive group of a p-ring is therefore separable whenever
it has an identity; actually, it must then be a bounded group.
Example 1.2. Let R be a torsion-free ring. If a 2 R is not nilpotent, then either t.a/ is idempotent
or all the types of a n .n 2 N/ are different.
Extending Ring to a Larger Group A more relevant problem is concerned with
the extension of the ring structure on a group to a larger group. The first question
which comes to mind is whether or not a ring can be extended to a ring on the
injective hull of its additive group. From (F) it is obvious that in the divisible hull of
a torsion group all products ought to be 0. Thus the real interest lies in the torsionfree
case, which can be dealt with satisfactorily.