Commutative ring/Sum and product of polynomial terms/Exercise

Let ${}R$ be a commutative ring, and ${}f,a_{i},b_{j}\in R$. Show the following equations:

$\sum _{i=0}^{n}a_{i}f^{i}+\sum _{j=0}^{m}b_{j}f^{j}=\sum _{k=0}^{\max(n,m)}(a_{k}+b_{k})f^{k}$

and

${\left(\sum _{i=0}^{n}a_{i}f^{i}\right)}\cdot {\left(\sum _{j=0}^{m}b_{j}f^{j}\right)}=\sum _{k=0}^{n+m}c_{k}f^{k}{\text{ where }}c_{k}=\sum _{r=0}^{k}a_{r}b_{k-r}.$

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