Theorem: the characteristic of a non unital integral domain must be $0$ or a prime.
Assume that $mn$ is the characteristic of a the integral domain for every $a$ in the integral domain we have $(mn)a^2=(ma)(na)=0$ then $ma$ or $na$ must be zero.
So for every $a$ in the integral domain $na$ OR $ma$ is zero but I can't say one of $m$ and $n$ works for every $a$ !