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 $k=\\gcd(a,n)$ 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 $\\gcd(c,n)=1$ , $c r$ 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 $\\gcd(w,n)=1$ such that $t=ws$ .

Note that $(hw)s=h(ws)=ht=t^{2}=(ws)^{2}=w^{2}s^{2}=w^{2}(gs)=(gw^{2})s$ . 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.∎

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 $\\gcd(a,n)$ .

单词	BehaviorExistsUniquelyfiniteCase
释义	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 $k=\\gcd(a,n)$ 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 $\\gcd(c,n)=1$ , $c r$ 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 $\\gcd(w,n)=1$ such that $t=ws$ . Note that $(hw)s=h(ws)=ht=t^{2}=(ws)^{2}=w^{2}s^{2}=w^{2}(gs)=(gw^{2})s$ . 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.∎ 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 $\\gcd(a,n)$ .
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