Let R be an infinite ring such that every subring of R different from {0} has a finite index in R. (By the index of a subring, we mean the index of its additive group in the additive group of R.) Prove that the additive group of R is cyclic.
L. Lovasz, J. Pelikan Ring Theoryfunctioninductionnumber theoryprime numberssuperior algebrasuperior algebra unsolved