  Torsion theories are a pinnacle in the theory of abelian categories. They are
a generalization of torsion abelian groups and in this generalization one of
the most studied is that whose torsionfree class consists of nonsingular
modules. To introduce the concept of singular interval we use the symmetric
idea of torsion theories, that is the torsion class determines the torsionfree
class and vice-versa, thus to introduce nonsingular intervals over an
upper-continuous modular complete lattice, (a.k.a idiom, a.k.a modular
preframe) we define the concept of \emph{division free} set. We introduce the
division free set of nonsingular intervals which defines a division set of
singular intervals in a canonical way. Several properties of division free sets
and some consequences of nonsingular intervals are explored allowing us to
develop a small part of a point-free nonsingular theory.
