Proof. Let $r$ be a generator of the additive group of $R$ . Then there exists $z \in \mathbb{Z}$ with $r^2=zr$ . If $z \ge 0$ , then $z$ is a behavior of $R$ . Assume $z<0$ . Note that $-z>0$ and $-r$ is also a generator of the additive group of $R$ . Since $(-r)^2=(-1)^2r^2=(-1)^2(zr)=(-z)(-r)$ , it follows that $-z$ is a behavior of $R$ . Thus, existence of behavior has been proven.

Let $a$ and $b$ be behaviors of $R$ . Then there exist generators $s$ and $t$ of the additive group of $R$ such that $s^2=as$ and $t^2=bt$ . If $s=t$ , then $as=s^2=t^2=bt=bs$ , causing $a=b$ . If $s \neq t$ , then it must be the case that $t=-s$ . (This follows from the fact that 1 and -1 are the only generators of $\mathbb{Z}$ .) Thus, $as=s^2=(-1)^2s^2=(-s)^2=t^2=bt=b(-s)=-bs$ , causing $a=-b$ . Since $a$ and $b$ are nonnegative, it follows that $a=b=0$ . Thus, uniqueness of behavior has been proven.