behavior exists uniquely (infinite case)

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


Let r be a generatorPlanetmathPlanetmathPlanetmath ( of the additive groupMathworldPlanetmath of R. Then there exists z with r2=zr. If z0, then z is a behavior of R. Assume z<0. Note that -z>0 and -r is also a generator of the additive group of R. Since (-r)2=(-1)2r2=(-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 s2=as and t2=bt. If s=t, then as=s2=t2=bt=bs, causing a=b. If st, then it must be the case that t=-s. (This follows from the fact that 1 and -1 are the only generators of .) Thus, as=s2=(-1)2s2=(-s)2=t2=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