New examples of indecomposable torsion-free abelian groups of finite rank and rings on them

The paper deals with new specific constructions of indecomposable torsion-free abelian groups of rank two and nonzero rings on them. They illustrate purely theoretical results and complement quite rare examples obtained during the classical as well as recent research of additive groups of rings. The presented results concerning the homogeneous groups remain true for groups of any finite nonzero rank. Moreover, the paper contains a construction of a torsion-free indecomposable abelian group of an arbitrary finite rank greater than two supporting an associative, but not commutative ring, as well as a ring which is neither associative nor commutative.