Let $G$ be a finite cyclic group of order $n$ with generator $g$. Given a ring $R$, then the group algebra over $R$ is isomorphic to $R[x] / (x^n - 1)$.
How to prove this simple fact? Surely the map $x \mapsto g$ contains $(x^n - 1)$ in its kernel, but how to show that they are equivalent. I thought some "dimension-type" argument would work, as if the kernel is "to big" we would have a different number of generators, but as the rank of a module is in general not unique (as written here) this argument will not work...
Hint: Use polynomial division by $x^n-1$, which is monic.