Consider the following definitions taken from the book Topics in Algebra by I. N. Herstein.
An integral domain $D$ is said to be of characteristic $0$ if the relation $ma=0$, where $a\neq 0$ is in $D$, and where $m$ is an integer, can hold only if $m=0$.
An integral domain $D$ is said to be finite characteristic if there exists a positive integer $m$ such that $ma=0$ for all $a\in D$.
I wish to prove that $D$ is of characteristic zero if and only if it is not of finite characteristic.
Suppose $D$ is of characteristic $0$. Then existence of positive integer $m$ such that $ma=0$ for all $a\in D$ clearly contradicts the required condition of characteristic $0$. (I presume there exists a nonzero element in $D$, although this is not done by Herstein.)
Conversely, suppose $D$ is not of finite characteristic. Then there exists no positive integer $m$ such that $ma=0$ for all $a\in D$. How do I proceed next?
My issue is that given the relation $ma=0$ where $0\ne a\in D$ is some fixed element, it may happen that $m'a=0$ for some positive $m'$, although that $m'$ is not "annihilating" all remaining elements and so is not contradicting the hypothesis. And yet that $m'$ is justifying why the ring is not of characteristic $0$.
One idea I had was to establish equivalent definitions using unity, and prove the result using that but Herstein does not endow his integral domains with unity.