behavior exists uniquely (infinite case)

The following is a proof that behavior exists uniquely for any infinite cyclic ring (http://planetmath.org/CyclicRing3) $R$.

Proof.

Let $r$ be a generator (http://planetmath.org/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 . Note that $-z>0$ and $-r$ is also a generator of the additive group of $R$. Since ${(-r)}^2={(-1)}^2{r}^2={(-1)}^2(zr)=(-z)(-r)$, 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 $s\ne t$, 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, $as=s^2={(-1)}^2{s}^2={(-s)}^2=t^2=bt=b(-s)=-bs$, causing $a=-b$. Since $a$ and $b$ are nonnegative, it follows that $a=b=0$. Thus, uniqueness of behavior has been proven.∎