This paper presents generalizations of results given in the book Geometry of Defining Relations in Groups by A. Yu. Ol'shanskii to the case of noncyclic torsion-free hyperbolic groups. In particular, it is proved that every noncyclic torsion-free hyperbolic group has a non-Abelian torsion-free quotient in which all proper subgroups are cyclic and the intersection of any two of them is nontrivial.