Let $R$ be a ring with $1$ and $R[x]$ be the polynomial ring in $x$ over $R$ with pointwise defined addition and convolution as multiplication.
Let $f\in R[x]$ be $$f_i:=\begin{cases}1\text{ if }i=1,\\0\text{ else }\end{cases}$$ then we can show by induction that $$(f^n)_i:=\begin{cases}1\text{ if }i=n,\\0\text{ else.}\end{cases}$$
Hello! This was given as an example in my linear algebra lecture on the topic of polynomial rings, however no proof was given and no name for this theorem or anything either. I assume however that this is somewhat popular and would be interested at looking at its proof and what it exactly does. Could anybody point me in the right direction?