Self-splitting Abelian groups

Abstract

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

Cotorsion theories and splitters

“…This interesting result, at that time motivated by studies on automorphism groups, received new support recently from investigations of cotorsion theories; see Salce [21] and Schultz [22]. Our paper will deal with Hausen's result, that is, with groups G such that Ext (G, G) = 0.…”

“…We will answer this question to the negative in Section 5. However, following Schultz [22], we first reduce the problem to the torsion-free case.…”

“…In order to investigate splitters, Schultz [22] introduced the very useful notion of a nucleus of a group. Definition 2.4.…”

“…Our paper will deal with Hausen's result, that is, with groups G such that Ext (G, G) = 0. Following Schultz [22], we call these modules splitters. Because of their importance, also other names are in use, see the 'Dictionary' on p. 351 in Ringel [17].…”

“…These self splitting modules are also called 'stones' by Kerner [15] (in contrast to 'bricks', which refers to the case that the endomorphism ring is a division ring, see [18]), exceptional modules by Rudakov [20], and Schur modules in Unger [24], see also [25]. We will stick to the name 'splitters' introduced by Schultz [22]. Our interest in splitters comes partly from their relatives, Whitehead groups, which have been investigated thoroughly over the last two decades; see results in [4] and in the more recent paper [1].…”