= Characteristic of an integral domain
{title2=$\operatorname{char}D=0\text{ or a prime}$}
The positive <characteristic of a ring> which is an <integral domain> is a <prime number>. The characteristic cannot be one because $1\ne0$. If its least positive value were $n=ab$ with $1<a,b<n$, minimality would make $a1$ and $b1$ nonzero, while $(a1)(b1)=n1=0$ would give <zero divisors>.
The analogous result holds for a nonzero commutative <nonunital ring> without <zero divisors>, using the <characteristic of a rng>. If $n=ab$ were composite and minimal, choose $r$ with $ar\ne0$. For every $s$, $(ar)(bs)=n(rs)=0$, so cancellation forces $bs=0$ for every $s$, contradicting minimality of $n$.
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