Abelian Groups - László Fuchs [Springer]

640 16 Endomorphism Rings
A ! A which must evidently be times of a map A ! A.) Consequently, the
matrix of admits a diagonal form with entries in Q.
This fact translated to maps asserts that there are automorphisms ; of A
(corresponding to A; B) such that D is represented by a diagonal matrix.
The zero columns in this matrix correspond to the kernel of ; consequently, the
kernel is of the form Ker D B 1 ˚˚B m with m n. Each component B j is
obtained as a sum of isomorphic copies of some of the components A i ,soitis
evidently quasi-isomorphic to the A i . Hence for Ker D Ker D Ker 1 D
.Ker/ we have
Ker D .B 1 ˚˚B m / D B 1 ˚˚B m :
(ii) Let i W A ! A i denote the projections in the given direct decomposition, and
let us fix quasi-isomorphisms ˛i W A i ! C .i D 1;:::;n/ where C is a pure
subgroup in A. Evidently, ˛i i C is an endomorphism of C, say, it is equal to
some i 2 D (Lemma 9.8 in Chapter 12). Let D gcdf 1 ;:::; n g be calculated
P
in D; thus, we have D 1 1 C C n n for suitable i 2 D. Then D
i i˛i i W A ! C is an endomorphism of A whose restriction to C acts like .
We use induction on n to complete the proof. If n D 1, thenC A 1 ,and
by purity equality holds. Next suppose n >1. Evidently, D is a nonzero
endomorphism of A that annihilates C. Owing to (i), Ker is a summand
of A, and also a direct sum of groups quasi-isomorphic to the A i .Since ¤ 0,
this summand has less than n components, so induction hypothesis applies. The
claim that C is a summand follows at once.
ut
Noetherian Endomorphism Rings It is a trivial observation that if End A
is noetherian, then A decomposes into the direct sum of a finite number of
indecomposable groups. It seems difficult to say much more in general about
groups with noetherian endomorphism rings: just consider arbitrarily large torsionfree
groups whose endomorphism rings are Š Z. The only hope is to put aside
torsion-free groups, and concentrate on torsion groups. Luckily, we have a complete
description in this case.
Theorem 5.6. Suppose A is a torsion group. End Aisleft.or right/ noetherian
exactly if A is finitely cogenerated.
Proof. If End A is noetherian, then A is a finite direct sum of indecomposable (i.e.
cocylic) groups, so it is finitely cogenerated. Conversely, if A is finitely cogenerated,
then by Theorem 5.3 in Chapter 4, A D B ˚ D with finite B and finite rank divisible
D. In the additive decomposition End A D Hom.B; D/ ˚ End D, the first summand
is finite, while the second summand is a finite direct sum of complete matrix rings
over the p-adic integers, for various primes. Thus End A is a finite extension of a
two-sided noetherian ring.
ut
Notice a kind of duality: a torsion group A has the maximum (minimum)
condition of subgroups if and only if End A has the minimum (maximum) condition
on left (or right) ideals.