ideal of elements with finite order

Theorem.  The set of all elements of a ring, which have a finite order in the additive groupMathworldPlanetmath of the ring, is a (two-sided) ideal of the ring.

Proof.  Let S be the set of the elements with finite order in the ring R.  Denote by o(x) the order of x.  Take arbitrary elements a,b of the set S.

If  lcm(o(a),o(b))=n=ko(a)=lo(b),  then


Thus  o(a-b)n<  and so  a-bS.

For any element r of R we have


Therefore, o(ra)o(a)<  and raS.  Similarly,  arS.

Since S satisfies the conditions for an ideal, the theorem has been proven.

Title ideal of elements with finite order
Canonical name IdealOfElementsWithFiniteOrder
Date of creation 2013-03-22 17:52:30
Last modified on 2013-03-22 17:52:30
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 20A05
Classification msc 16D25
Related topic OrderGroup
Related topic Lcm
Related topic Multiple
Related topic OrdersOfElementsInIntegralDomain
Related topic CharacteristicOfFiniteRing