Example for a functor $F:Mod(A)\rightarrow Mod(B)$

Question

I'm trying to find a simple example for a functor $F$ between modules, which is non linear and I don't really know how to approach this. Intuitively, I'm thinking matrix modules. A hint would be appreciated.

Answer 1

As I said in the comments, a trivial example is given by any nonzero constant functor.

The proof is easy, and you gave it in the comments.

Another, less trivial (and actually very interesting in some instances) example is given by the "free module functor". Indeed, for any $A$-module $M$, you can forget that it's an $A$-module and get a set $UM$, and then you can take the free $B$-module on that: $B[UM]$.

This is also wildly non linear (in this free module, letting $[m]$ denote the generator associated to $m\in M$, $[m]+[n]\neq [m+n]$)

Using the forgetful functor to $\mathbf{Set}$ followed by some functor $\mathbf{Set}\to Mod(B)$ will easily produce tons of other examples.