characteristic of finite ring

The characteristicPlanetmathPlanetmath ( of the residue class ring /m, which contains m elements, is m, too.  More generally, one has the

Theorem.  The characteristic of a finite ring divides the number of the elements of the ring.

Proof. Let n be the characteristic of the ring R with m elements.  Since m is the order ( of the group  (R,+),  the Lagrange’s theorem implies that

ma= 0aR.

Let  m=qn+r  where  0r<n.  Because

ra=(m-qn)a=ma-q(na)= 0-0= 0aR

and n is the least positive integer ν making all  νa=0,  the number r must vanish.  Therefore,  m=qn,  i.e.  nm.

Remark.  A ring R, the polynomial ring R[X] and the ring R[[X]] of formal power series have always the same characteristic.

Title characteristic of finite ring
Canonical name CharacteristicOfFiniteRing
Date of creation 2013-03-22 19:10:19
Last modified on 2013-03-22 19:10:19
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 16B99
Related topic MultipleMathworldPlanetmathPlanetmath
Related topic IdealOfElementsWithFiniteOrder