Es sei ${\displaystyle {}V}$ ein Modul über dem kommutativen Ring ${\displaystyle {}R}$. Es seien ${\displaystyle {}s_{1},\ldots ,s_{k}\in R}$ und ${\displaystyle {}v_{1},\ldots ,v_{n}\in V}$. Zeige

${\displaystyle {}{\left(\sum _{i=1}^{k}s_{i}\right)}\cdot {\left(\sum _{j=1}^{n}v_{j}\right)}=\sum _{1\leq i\leq k,\,1\leq j\leq n}s_{i}\cdot v_{j}\,.}$