Restricted-finite groups with some applications in group rings

We carry out a study of groups G in which the index of any infinite subgroup is finite. We call them restricted-finite groups and characterize finitely generated not torsion restricted-finite groups. We show that every infinite restricted-finite abelian group is isomorphic to Z× K or Zp^∞× K, where K is a finite group and p is a prime number. We also prove that a group G is infinitely generated restricted-finite if and only if $G = AT$ where A and T are subgroups of G such that A is normal quasi-cyclic and T is finite. As an application of our results, we show that if G is not torsion with finite $G'$ and the group-ring $RG$ has restricted minimum condition, then R is a semisimple ring and G T, where T is finite whose order is unit in R. The converse is also true with certain conditions including G = T× Z.