Vol. 27, No. 3, 1968

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Sums of automorphisms of a primary abelian group

Frank Castagna

Vol. 27 (1968), No. 3, 463–473

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.

Mathematical Subject Classification
Primary: 20.30
Received: 15 January 1968
Published: 1 December 1968
Frank Castagna