Linear Algebra over a Ring/Modules and linear functions

Exercises

 * 1) Prove that for a function $$f: M \to N$$ between left $$R$$-modules, the following are equivalent:
 * 2) $$f$$ is linear
 * 3) For all $$m,n \in M$$ and $$r \in R$$, we have $$f(m + n) = f(m) + f(n)$$ and $$f(rm) = rf(m)$$
 * 4) For all $$m,n \in M$$ and $$r \in R$$, we have $$f(m + rn) = f(m) + rf(n)$$
 * 5) For all $$m,n \in M$$ and $$r,s \in R$$, we have $$f(rm + sn) = rf(m) + sf(n)$$