# Preradical

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In [mathematics](/source/Mathematics), a **preradical** is a [subfunctor](/source/Subfunctor) of the [identity functor](/source/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

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