= Solution
For a <ring> $R$, its <ring characteristic> is the least positive <integer> $n$ with $n1_R=0$, or zero if no positive such integer exists. Equivalently it is the least positive integer satisfying $nr=0$ for every $r\in R$, because $nr=(n1_R)r$. This additive formulation also defines the <characteristic of a rng> when an identity is not assumed.
Let $R$ be an <integral domain>. Since $1_R\ne0$, its characteristic is not one. If its positive characteristic were a composite integer $n=ab$ with $1<a,b<n$, minimality would give $a1_R\ne0$ and $b1_R\ne0$, but
$$
(a1_R)(b1_R)=n1_R=0.
$$
This contradicts the absence of <zero divisors>. Therefore
$$
\boxed{\operatorname{char}R=0\quad\text{or a prime number}.}
$$
This is the <characteristic of an integral domain> theorem. If an integral domain is defined without requiring an identity, the same conclusion follows from the additive definition: for a hypothetical composite characteristic $n=ab$, choose $r$ with $ar\ne0$, which exists by minimality of $n$. Then $(ar)(bs)=n(rs)=0$ for every $s$, forcing $bs=0$ for every $s$ and contradicting minimality again.
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