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This paper is concerned with
the problem: For which abelian groups G does the group of automorphisms of
G generate the ring of endomorphisms of G? R. S. Pierce has shown that
if G is a 2-primary group, all of whose finite Ulm invariants are equal to
one, then the subring of E(G) generated by the group of automorphisms
of G is properly contained in E(G). The groups considered in this paper
are p-primary abelian groups where p is a fixed prime number greater than
two. The paper is divided into four parts. In §1 the following theorem is
proved:
Theorem. If G is a countable reduced p-primary, (p > 2), abelian group, then
every endomorphism of G is a sum of two automorphisms.
The second part gives an extension of this theorem to the case where G is the
direct sum of such groups. In §3 the result of this theorem is established for torsion
complete abelian groups, using some known results about their endomorphism rings.
Finally an example is given (for an arbitrary prime p) of a reduced p-primary abelian
group G for which there are endomorphisms in E(G) that are not sums of
automorphisms.
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