Abelian Groups - László Fuchs [Springer]

2 Rings on Groups 679
is a subring, and N D A \ N is the largest nilpotent ideal of A; they satisfy
Q ˝ S D S ; Q ˝ N D N .
Furthermore, S ˚ N is a subring of finite index in A.
Proof. Everything follows without difficulty from the classical Wedderburn theorem,
except for the last statement. That S ˚ N is a finite index subring requires a
long proof. (We can add that the additive group of S is quotient-divisible.) ut
Role of Basic Subgroups We turn our attention to an interesting phenomenon
underlining the relevance of basic subgroups.
Proposition 2.8 (Fuchs [5]). Let A be a p-group, and B D ˚i2I hb i i a basic
subgroup of A. Any multiplication 2 Mult A is completely determined by the
set of the .b i ; b j / for all i; j 2 I.
Conversely, every choice of the elements .b i ; b j / for all i; j 2 I .subject to the
necessary condition that o..b i ; b j // minfo.b i /; o.b j /g/ gives rise to a naring
on A.
Proof. There are several proofs to choose from, the following is most elementary
and direct. Needless to say, that if is given for the basis elements of B (of course,
the values need not be in B), then it extends linearly to all of B. Tofind.a; a 0 / for
all a; a 2 A,leto.a 0 / D p k , and write accordingly a D b C p k x with b 2 B; x 2 A,to
obtain
.a; a 0 / D .b; a 0 / C .p k x; a 0 / D .b; a 0 / C .x; p k a 0 / D .b; a 0 /:
In a similar fashion, .b; a 0 / D .b; b 0 / for a suitable b 0 2 B, establishing the first
part of our claim.
For the second part, note that, every b 2 B being a unique linear combination
of the b i , the ring postulates for .b; b 0 / for all b; b 0 2 B are readily checked. If
is extended to the whole of A (in the way shown in the preceding paragraph), then
just a routine verification is needed to show that the extended is well defined and
satisfies the requisite postulates.
ut
Example 2.9. Let B be the torsion-completion of the direct sum B D˚i2I hb i i of cyclic p-groups.
Rings can be defined on B in various ways, we need to specify the products b i b j .
(a) Putting b i b j D b i or b j according as o.b i / o.b j / or not.
(b) Order the index set I, and in case o.b i / D o.b j /,setb i b j D b i or D b j according as i j or
i > j.
(c) Of course, b i b j can be chosen arbitrarily, e.g. b i b j D b k at random (subject to the necessary
order condition) or a linear combination.
In (a)–(c), B is a subring of the ring on B such that all the products belong to B. In case (b)–(c),
associativity is in doubt.
Example 2.10. Let A be a p-group with basic subgroup B. Then Mult A Š Hom.B ˝ B; A/; this
follows from Proposition 2.8.
Absolute Properties When dealing with rings on a given group A, an inevitable
problem is to find those subgroups of A that enjoy certain property P, like being an
ideal, or an annihilator, in every ring supported by A. We shall call them absolute P
on A.