Let $R$ be a finite commutative ring with multiplicative identity $1$. Let $S \subset R$ be the set of elements with multiplicative inverses in $R$. Assume that $S$ consists of $k$ elements, and that $R$ consists of $n$ elements. Does $s^k=1$ for all $s\in S$?
I attempted to solve this problem by imagining $R=\mathbb Z_n$ and $S=\mathbb Z_n\backslash\{a:gcd(a,n)\ne1\}$. Then it follows pretty easily since $|S|=k=\phi(n)$. Therefore by Fermat's Little Theorem we have $s^k\equiv1 \pmod n$ for all $s \in S$.
I am not sure how to generalize this into a proof, however. Does this work since any $R$ is going to be isomorphic to $\mathbb Z_n$?
Any help would be much appreciated!