Preradical

Short description: Subfunctor in mathematics

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 [math]\displaystyle{ \sigma }[/math] and [math]\displaystyle{ \tau }[/math], [math]\displaystyle{ \sigma\leq\tau }[/math], if for any left R-module M, [math]\displaystyle{ \sigma M\leq \tau M }[/math]. 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