A simple solution of Stratton and Webb’s problem

Abstract

The paper contains a simple solution of Stratton and Webb’s problem related to the so-called absolute annihilators of abelian groups researched also by Feigelstock. Namely, there are provided constructions of indecomposable torsion-free abelian groups G of any finite rank greater than two for which the quotient group modulo the subgroup generated by squares of all possible rings on G supports a non-zero associative ring structure. They are much less complicated than those obtained before and show that groups satisfying the mentioned condition are not as rare phenomena as might be supposed. Moreover, most of these constructions led to new examples of CR- and AR-groups, i.e. abelian groups supporting only commutative and only associative rings, respectively. In particular, the first examples of indecomposable as well as decomposable AR-groups which do not satisfy the condition CR are presented. The first example of an indecomposable torsion-free group satisfying the condition CR only in the class of associative rings is also obtained.