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.

References

• Stenstrom, Bo Rings of Quotients: An Introduction To Methods Of Ring Theory – Chapter 6, Springer, ISBN 0387071172
• Bican, L., Kepka, T. and Nemec, P. Rings, Modules, and Preradicals, Lecture Notes in Pure and Applied Mathematics, M. Dekker, 1982, ISBN 0824715683