A point-free version of torsionfree classes and the Goldie torsion theory

Abstract: 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.