# 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 $k=\mathrm{gcd}(a,n)$ and $b\in \mathbb{Z}$ with $a=bk$. Since $\mathrm{gcd}(b,n)=1$, there exists $c\in \mathbb{Z}$ with $bc\equiv 1\mathrm{mod}n$. Since $\mathrm{gcd}(c,n)=1$, $cr$ 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 $\mathrm{gcd}(w,n)=1$ 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 $\mathrm{gcd}(w,n)=1$. Therefore, $gw\equiv h\mathrm{mod}n$. Since $g$ and $h$ are both positive divisors of $n$ and $\mathrm{gcd}(w,n)=1$, it follows that $g=\mathrm{gcd}(g,n)=\mathrm{gcd}(gw,n)=\mathrm{gcd}(h,n)=h$. 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 $\mathrm{gcd}(a,n)$.

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

Canonical name | BehaviorExistsUniquelyfiniteCase |

Date of creation | 2013-03-22 16:02:35 |

Last modified on | 2013-03-22 16:02:35 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 13 |

Author | Wkbj79 (1863) |

Entry type | Proof |

Classification | msc 16U99 |

Classification | msc 13M05 |

Classification | msc 13A99 |