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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 ${\displaystyle \sigma }$ and ${\displaystyle \tau }$, ${\displaystyle \sigma \leq \tau }$, if for any left R-module M, ${\displaystyle \sigma M\leq \tau M}$. 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

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