2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/215557We Classify the rational quadratic extensions K and the finite groups G for which the group ring R[G] of G over the ring R of integers of K has the property that the group of units of augmentation 1 of R[G] is hyperbolic. We also construct units in a non-split quaternion algebra over R.15 pages, this work is part of the PHd. Thesis of the third author. The paper was accepted in Proceedings Mathematical SciencesRings and AlgebrasGroup Theory16U60, 16S34, 20F67text