I'm finding the algebra behind cyclic codes somewhat tricky. The starting point is easy enough: $C\subseteq \mathbb F\_q^n$ is cyclic if any cyclic shift of a codeword $c\in \mathbb F\_q^n$ is still in $C$. Then I got hit with this: cyclic codes correspond to the ideals of
$$\mathbb F\_q[x]/(x^n-1). $$
Now, I have some background in abstract algebra, mostly from group theory. I can recognize a ring and a quotient, but I'm having trouble to see the equivalence. Can anyone explain it to me in *very simple* terms?