On a theorem of Stelzer for some classes of mixed groups

Abstract

We identify some classes $\mathcal{C}$ of mixed groups such that if $G\in \mathcal{C}$ has the cancellation property then the Walk-endomorphism ring of $G$ has the unit lifting property. In particular, if $G$ is a self-small group of torsion-free rank at most $4$ with the cancellation property then it has a decomposition $G=F\oplus H$ such that $F$ is free and the Walk-endomorphism ring of $H$ has the unit lifting property.