behavior exists uniquely (finite case)

The following is a proof that behavior exists uniquely for any finite cyclic ring $R$.

Proof.

Let $n$ be the order (http://planetmath.org/OrderRing) of $R$ and $r$ be a generator (http://planetmath.org/Generator) of the additive group of $R$. Then there exists $a\in \mathbb{Z}$ with $r^2=ar$. Let $k=\gcd (a,n)$ and $b\in \mathbb{Z}$ with $a=bk$. Since $\gcd (b,n)=1$, there exists $c\in \mathbb{Z}$ with $bc\equiv 1\mathrm{mod}n$. Since $\gcd (c,n)=1$, $cr$ is a generator of the additive group of $R$. Since ${(cr)}^2=c^2{r}^2=c^2(ar)=c^2(bkr)=c(bc)(kr)=k(cr)$, it follows that $k$ is a behavior of $R$. Thus, existence of behavior has been proven.

Let $g$ and $h$ be behaviors of $R$. Then there exist generators $s$ and $t$ of the additive group of $R$ such that $s^2=gs$ and $t^2=ht$. Since $t$ is a generator of the additive group of $R$, there exists $w\in \mathbb{Z}$ with $\gcd (w,n)=1$ such that $t=ws$.

Note that $(hw)s=h(ws)=ht=t^2={(ws)}^2=w^2{s}^2=w^2(gs)=(g{w}^2)s$. Thus, $g{w}^2\equiv hw\mathrm{mod}n$. Recall that $\gcd (w,n)=1$. Therefore, $gw\equiv h\mathrm{mod}n$. Since $g$ and $h$ are both positive divisors of $n$ and $\gcd (w,n)=1$, it follows that $g=\gcd (g,n)=\gcd (gw,n)=\gcd (h,n)=h$. Thus, uniqueness of behavior has been proven.∎

Note that it has also been shown that, if $R$ is a finite cyclic ring of order $n$, $r$ is a generator of the additive group of $R$, and $a\in \mathbb{Z}$ with $r^2=ar$, then the behavior of $R$ is $\gcd (a,n)$.