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

$$ma=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}\forall a\in R.$$

Let  $m=qn+r$  where  .  Because

$$ra=(m-qn)a=ma-q(na)=\mathrm{\hspace{0.33em}0}-0=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}\forall a\in R$$

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.