Question: Suppose that $R$ is a commutative ring without zero-divisors. Show that the characteristic of $R$ is either $0$ or prime.
I have established that every element in a commutative ring $R$ without zero divisors have the same additive order $n$.
Now, if no such additive order n exists, then the characteristic of $R$ is $0$.
Obviously, if a finite additive order exists, Char of $R$ is finite. How do I show that Char of $R$ is prime? It probably involves lagrange's theorem and the order of the element in $R$.
Hint is appreciated.
Thanks in advance.