Abelian groups in which endomorphic images are fully invariant

Abstract

In this paper we concern ourselves with the following problem due to Szele and Szendrei [lo] and listed as Problem 47 in [4] : Is the endomorphism ring commutative if every endomorphic image is fully invariant ? In [8], Reid gave a counterexample to the Szele-Szendrei problem by constructing a ring of rank 4 with specified properties to guarantee, by A. L. S. Corner’s excellent work [3], a torsion free group of rank 8 for which endomorphic images are fully invariant but whose endomorphism ring is isomorphic to the constructed non-commutative ring. The investigation here will show that for every positive integer n there are torsion free counterexample groups of rank 8n with endomorphism rings of rank 4n. We will also indicate to what extent this list is exhaustive and show that there are no counterexamples of a specified type of rank less than 8. In Section 2 we discuss preliminary concepts (particularly stability and n-stability for abelian groups and rings) and restrict our attention to torsion free abelian groups of finite rank. In Section 3 we indicate the sufficiency of studying strongly indecomposable groups and n-stable subrings of division rings. In Section 4 we characterize all n-stable subrings of the rational quaternions which generalize to give the counterexamples of rank gn found in Section 5. Finally, in Section 6, we point out that 8 is the minimum rank of groups having endomorphism rings isomorphic to n-stable non-commutative subrings of the quaternions.