orders of elements in integral domain


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

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


Thus, because  a0  and there are no zero divisors, an equationna=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
Canonical name OrdersOfElementsInIntegralDomain
Date of creation 2013-03-22 15:40:28
Last modified on 2013-03-22 15:40:28
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Theorem
Classification msc 13G05
Related topic OrderGroup
Related topic IdealOfElementsWithFiniteOrder