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.
Best Answer
For rings not necessarily with identity, you can define the characteristic as the non negative generator of the ideal in $\mathbb{Z}$ formed by the integers $n$ such that $nr=0$, for all $r\in R$.
Suppose this ideal is not $\{0\}$; then it is $k\mathbb{Z}$, with $k>0$. Suppose $k$ is not prime, so $k=ab$, with $0<a<k$ and $0<b<k$.
By definition of $k$, there are $r\in R$ and $s\in R$ with $$ ar\ne0,\qquad bs\ne0 $$