Preradical

In mathematics, a preradical is a subfunctor of the identity functor in the category of left modules over a ring with identity. The class of all preradicals over R-mod is denoted by R-pr. There is a natural order in R-pr given by, for any two preradicals $$ \sigma $$ and $$ \tau $$, $$\sigma\leq\tau$$, if for any left R-module M, $$\sigma M\leq \tau M$$. With this order R-pr becomes a big lattice.