Proof. Let $a$ be arbitrary non-zero element. Any multiple $na$ may be written as $$na = n(1a) = \underbrace{1a+1a+\cdots+1a}_{n} = (\underbrace{1+1+\cdots+1}_{n})a = (n1)a.$$ Thus, because $a \ne 0$ and there are no zero divisors, an equation $na = 0$ is equivalent with the equation $n1 = 0$ . So $a$ must have the same order as the unity of $D$ .

Note. The order of the unity element is the characteristic of the integral domain, which is 0 or a positive prime number.