# 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. ∎

Title | behavior exists uniquely (infinite case) |
---|---|

Canonical name | BehaviorExistsUniquelyinfiniteCase |

Date of creation | 2013-03-22 16:02:32 |

Last modified on | 2013-03-22 16:02:32 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 11 |

Author | Wkbj79 (1863) |

Entry type | Proof |

Classification | msc 13A99 |

Classification | msc 16U99 |