R-modules anyone?

[battletoad]

Cant seem to fathom what an R-module is. Thing is i can prove if something (elementary structures for now) is an R-module or not, but i dont get what the basic structure 'looks' like. Let me explain...

 

Graphs and rings are fine, in being sets which satisfy certain criteria. So i understand that they are special sets, operative word SETS, a structure i understand. R-modules on the other hand utilise these special sets (abelian groups, rings), uses an endomorphism onto, say, the group 'space', then end up being a structure i vaguely understand (obviously a set itself, but different somehow to me). So for now i view an R-module as a ring (again, satisfying certain axioms) but il ultimately would like to know what it 'looks like'. Hope im making myself understood.

 

All in all, im thinking i should move away from my need to familiarise myself with mathematical structures, ideas, concepts and so forth, only deal with them on a definition basis. Perhaps then il go into middle age with a full head of hair :/

[ajb]

I think of R-modules (Modules over a ring, but we can have modules over other things like algebras etc) as "vector spaces in which the scalars are now in the ring".

 

You have the additive group structure and a product between the elements of the module and elements of the ring.

 

Good geometric example includes the ring of smooth functions on a (super)manifold and the module of vector fields.

 

Your question of "how things look like" is best answered in category theory where one places the emphasis on the morphisms rather than the elements.

 

Morphisms are understood as "structure preserving maps". So, fix a ring. Then if [math]M[/math] and [math]N[/math] are modules over this ring then a morphism of R-modules is a map

 

[math]\phi: M \rightarrow N[/math],

 

(as sets) such that

 

[math]\phi(rm + sn) = r \phi(m) + s \phi(n)[/math]

 

where [math]r,s[/math] are elements of the ring and [math]m,n[/math] are in [math]M[/math].

 

With these morphisms we have the category of R-modules.

[battletoad]

Thanks for the heads up. Ur 'vector spaces where the scalars are now in the ring' (tbh thought that same thing but maybe i needed time to fully understand it all) fits in nicely with what im doing in group theory atm, automorphisms and such.

Planning to read up on vector spaces again forgot most of it...would've been great if i didnt sell that textbook...