orders of elements in integral domain

Theorem.

Let  $(D,+,\cdot )$  be an integral domain, i.e. a commutative ring with non-zero unity 1 and no zero divisors.  All non-zero elements of $D$ have the same order (http://planetmath.org/OrderGroup) in the additive group  $(D,+)$.

Proof.  Let $a$ be arbitrary non-zero element.  Any multiple (http://planetmath.org/GeneralAssociativity) $na$ may be written as

$$na=n(1a)=\underset{n}{\underbrace{1a+1a+\mathrm{\cdots }+1a}}=(\underset{n}{\underbrace{1+1+\mathrm{\cdots }+1}})a=(n1)a.$$

Thus, because  $a\ne 0$  and there are no zero divisors, an equation  $na=0$  is equivalent (http://planetmath.org/Equivalent3) with the equation  $n1=0$.  So $a$ must have the same as the unity of $D$.

Note.  The of the unity element is the characteristic (http://planetmath.org/Characteristic) of the integral domain, which is 0 or a positive prime number.