# characteristic of finite ring

The characteristic  (http://planetmath.org/Characteristic) of the residue class ring $\mathbb{Z}/m\mathbb{Z}$, 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 (http://planetmath.org/OrderGroup) of the group  $(R,\,+)$,  the Lagrange’s theorem implies that

 $ma\;=\;0\quad\forall a\in R.$

Let  $m=qn\!+\!r$  where  $0\leqq r.  Because

 $ra\;=\;(m\!-\!qn)a\;=\;ma\!-\!q(na)\;=\;0\!-\!0\;=\;0\quad\forall a\in R$

and $n$ is the least positive integer $\nu$ making all  $\nu a=0$,  the number $r$ must vanish.  Therefore,  $m\,=\,qn$,  i.e.  $n\mid m$.

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 CharacteristicOfFiniteRing 2013-03-22 19:10:19 2013-03-22 19:10:19 pahio (2872) pahio (2872) 7 pahio (2872) Theorem msc 16B99 Multiple   IdealOfElementsWithFiniteOrder