Proof. Let $n$ be the order of $R$ and $r$ be a 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 $bc \equiv 1 \operatorname{mod} n$ . Since $\gcd(c,n)=1$ , $cr$ is a generator of the additive group of $R$ . Since $(cr)^2=c^2r^2=c^2(ar)=c^2(bkr)=c(bc)(kr)=k(cr)$ , 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 $(hw)s=h(ws)=ht=t^2=(ws)^2=w^2s^2=w^2(gs)=(gw^2)s$ . Thus, $gw^2 \equiv hw \operatorname{mod} n$ . Recall that $\gcd(w,n)=1$ . Therefore, $gw \equiv h \operatorname{mod} n$ . 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.