A note on Feigelstock's conjecture on the equivalence of the notions of nil and associative nil groups in the context of additive groups of rings of finite rank

Abstract

In the class of torsion-free abelian groups of finite rank, Feigelstock's conjecture on the equivalence of the notions of nil and associative nil groups is reduced to indecomposable groups. Torsion-free abelian groups $A$ of rank two such that every associative ring on $A$ is commutative but there exists a~non-commutative ring with the additive group $A$, are classified. Moreover, several valuable results concerning rings on torsion-free abelian groups of rank two achieved by Beaumont, Wisner, Jackett, Aghdam and Najafizadeh are complemented and their proof are greatly simplified.$