On the inverse operations in the class of preradicals of a module category, I

In the class $\mathbb{PR}$ of preradicals of the category of left $R$-modules $R$-Mod a new operation is defined and studied, namely the left quotient with respect to join. Some properties of this operation are shown, its compatibility with the lattice operations of $\mathbb{PR}$ (meet and join of preradicals), as well as the relations with some constructions in the “big” lattice $\mathbb{PR}$. Also some particular cases are examined.