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§0. Introduction. Seotion 1
of the paper deals with the structure of the left End(T)-module Ext(F,T) where F is
a torsion-free group and T a torsion group. The main theorem states that if the rank
of F is n then the rank of Ext(F,T) over End (T) is at most n. It is also shown that
Ext(F,T) is a torsion End (T)-module.
In §2 the case when F is a torsion-free divisible group of finite rank is considered.
It is shown that ℱm, the elements of the End (T)-End (F) bimodule Ext(F,T) whose
splitting length is at most m, are a sub-bimodule. Moreover ℱ = ⋃
i=1∞ℱi as well
as ℱm are not finitely generated sub-bimodules. However, when T is the
direct sum of cyclic groups, every finite number of elements of ℱ can be
embedded in a submodule generated by at most rank F elements belonging to
ℱm.
We introduoe some notation. The word “group” will always mean abelian
group. Q will stand for the additive group of rational numbers, N the natural
numbers, and N0 the nonnegative integers. Let G be a group, then T(G)
is the maximal torsion subgroup of G. For x ∈ G then hp(x) will denote
its generalized p-height in G,o(x) its order, and ⟨x⟩ the cyclio subgroup
generated by x. For a torsionfree element x ∈ G,U(x), the Ulm matrix of x, will
refer to the matrix hp(pix) over all i ∈ N0 and primes p[9]. For the module
structure of Ext(F,T) over End (T)-End (F) we refer the reader to [8].
Following [8] for α ∈ End(T)[γ ∈ End(F)] and E ∈ Ext(F,T) by αE[Eγ] we
mean the extension arising from the pushout [pullback] diagram of E along
α[γ].
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