Additive groups of zero square rings

Summary

We study abelian groups <Emphasis Type=”Italic”>G</Emphasis> that are additive groups of rings <Emphasis Type=”Italic”>R </Emphasis>such that <Emphasis Type=”Italic”>x</Emphasis><Superscript>2</Superscript>=0 for all <Emphasis Type=”Italic”>x </Emphasis>&isin; <Emphasis Type=”Italic”>R</Emphasis>, but <Emphasis Type=”Italic”>RR</Emphasis> &ne; {0}. We call such groups <Emphasis Type=”Italic”>G</Emphasis> zero square groups. If <Emphasis Type=”Italic”>G</Emphasis> is not the additive group of such a ring, we call <Emphasis Type=”Italic”>G</Emphasis> a non-zero square group. We determine the non-zero square groups among the torsion, divisible, and completely decomposable abelian groups. We construct zero square groups with (essentially) prescribed endomorphism rings. As a consequence, there exist many torsion-free strongly indecomposable zero square groups of finite rank.