Let be a symmetric space of noncompact type. A result of Gelander provides exponential upper bounds in terms of the volume for the torsion homology of the noncompact arithmetic locally symmetric spaces . We show that under suitable assumptions on this result can be extended to the case of nonuniform arithmetic lattices that may contain torsion. Using recent work of Calegari and Venkatesh we deduce from this upper bounds (in terms of the discriminant) for of the ring of integers of totally imaginary number fields . More generally, we obtain such bounds for rings of -integers in F.