A ring with prime characteristic

Question

Let $p$ be a prime and let $R$ be a commutative ring with characteristic $p$. Prove that the number of elements of the set $$S_k=\{x\in R\;\lvert \;x^p=k\}\quad \text{for} \quad k\in \{1,2,\dots,p\}$$ is same for each $k$.

This was a problem in my exam. No one in my class solved it. I have no idea for it. How to proceed? Can someone help?

Answer 1

Well, $x^p=k$ is equivalent to $(x-k)^p=0$. Hence $x \mapsto x-k$ gives $S_k \cong S_0$.

Answer 2

Let's start observing that $k\in S_k$, so that $S_k\neq\emptyset$.

Given $k\neq k^\prime$ pick any $x\in S_{k^\prime-k}$. Now the application $$ z\mapsto z+x $$ gives a bijection $S_k\simeq S_{k^\prime}$ because in characteristic $p$ the map $y\mapsto y^p$ is a ring homomorphism.