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:
   1. $f$ is linear
   2. For all $m,n\in M$ and $r\in R$, we have $f(m+n)=f(m)+f(n)$ and $f(rm)=rf(m)$
   3. For all $m,n\in M$ and $r\in R$, we have $f(m+rn)=f(m)+rf(n)$
   4. For all $m,n\in M$ and $r,s\in R$, we have $f(rm+sn)=rf(m)+sf(n)$