Let $R$ be a ring with $1$, $f\in R[x]$ and $f_i=0$ for all $i>n$. Show that $$f=\sum_{j=0}^{n}f_jx^j.$$
Show furthermore that if $f=\sum_{j=0}^{n}g_jx^j$, we have $g_i=f_i$ for $i=0,...,n$.
I'm currently studying linear algebra and a bit confused about this exercise. As far as I can tell the first equation given is just a standard representation of a polynomial over a ring and is defined as such. Am I wrong? What is there to prove?