# 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

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

Thus, because  $a\neq 0$  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.

Title orders of elements in integral domain OrdersOfElementsInIntegralDomain 2013-03-22 15:40:28 2013-03-22 15:40:28 pahio (2872) pahio (2872) 9 pahio (2872) Theorem msc 13G05 OrderGroup IdealOfElementsWithFiniteOrder