ModulFunktoren

ModulFunktoren, a term originating from abstract algebra, refers to a specific type of functor between categories of modules. In essence, a ModulFunktor maps modules over a ring R to modules over a ring S, while also preserving the module structure and respecting the homomorphisms between modules. This mapping is not arbitrary; it is governed by specific rules that ensure the integrity of the module category under the transformation.

To be precise, a ModulFunktor F from the category R-Mod to the category S-Mod is a mapping that assigns to each R-module M an S-module F(M), and to each R-homomorphism f: M -> N a corresponding S-homomorphism F(f): F(M) -> F(N). These assignments must satisfy two fundamental properties: the functorial identity and the functorial composition. The identity property states that for the zero homomorphism 0: M -> M, its image F(0) must be the zero homomorphism on F(M). The composition property requires that for any two R-homomorphisms f: M -> N and g: N -> P, the functor applied to their composition must equal the composition of the functor applied to each: F(g o f) = F(g) o F(f).

The concept of ModulFunktoren is crucial in understanding deeper relationships between different module categories. They provide a systematic way to transfer properties and structures from one module setting to another, aiding in the classification and study of modules. Examples include Hom functors and tensor product functors, which are fundamental tools in homological algebra and representation theory.