behavior exists uniquely (finite case)

The following is a proof that behavior exists uniquely for any finite cyclic ring R.


Let n be the order ( of R and r be a generatorPlanetmathPlanetmathPlanetmath ( of the additive groupMathworldPlanetmath of R. Then there exists a with r2=ar. Let k=gcd(a,n) and b with a=bk. Since gcd(b,n)=1, there exists c with bc1modn. Since gcd(c,n)=1, cr is a generator of the additive group of R. Since (cr)2=c2r2=c2(ar)=c2(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 s2=gs and t2=ht. Since t is a generator of the additive group of R, there exists w with gcd(w,n)=1 such that t=ws.

Note that (hw)s=h(ws)=ht=t2=(ws)2=w2s2=w2(gs)=(gw2)s. Thus, gw2hwmodn. Recall that gcd(w,n)=1. Therefore, gwhmodn. 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 with r2=ar, then the behavior of R is 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