Abelian Groups - László Fuchs [Springer]

1 Endomorphism Rings 621
Recently, several publications deal with the so-called algebraic entropy which was recently
introduced in abelian groups. The paper Dikranjan–Goldsmith–Salce–Zanardo [1] contains lots of
interesting results on the entropy of endomorphisms.
Exercises
(1) If jAj Dp n ,thenj End Aj p n2 .
(2) (a) If G A, then Hom.A; G/ is a right ideal in End A.
(b) If G is fully invariant, then Hom.A; G/ is a two-sided ideal.
(3) (Lawver) All endomorphic images of A are fully invariant exactly if, for every
a 2 A and for all ; 2 End A,thereisab 2 A such that ./a D b.
(4) Show that the finite topology of End A for a separable torsion-free group A can
be defined intrinsically (i.e. without reference to A).
(5) Describe the finite topology of J p as an endomorphism ring of Z.p 1 / and as
that of J p .
(6) (a) The direct sum and the direct product of elementary p-groups T p for
different primes p have isomorphic endomorphism rings.
(b) However, these endomorphism rings are not isomorphic as topological
rings (equipped with the finite topology).
(7) For an infinite group A, EndA is always infinite. Give examples where jAj <
j End Aj,andwherejAj > j End Aj.
(8) (a) Let fG i g i2I be a system of subgroups of A which is directed upwards
under inclusion such that [ i2I G i D A. Define a topology in End A by
declaring the set of left ideals L i Df 2 End A j G i D 0g as a base
of neighborhoods about 0. Show that End A is a complete group in this
topology, and if the G i are fully invariant in A, then End A is a topological
ring.
(b) (Pierce) Let A be a p-group and G i D AŒp i .i 0.
(12) Suppose A D˚i2I A i with countable summands A i . In the matrix representations
of endomorphisms, every column contains at most countably many
non-zero entries.
(13) The set End E A of E-endomorphisms of A is the center of the ring E D End A.
(14) For any group A, the center of the ring End.A ˚ Z/ is Š Z.