behavior exists uniquely (infinite case)
The following is a proof that behavior exists uniquely for any infinite cyclic ring (http://planetmath.org/CyclicRing3) .
Let be a generator (http://planetmath.org/Generator) of the additive group of . Then there exists with . If , then is a behavior of . Assume . Note that and is also a generator of the additive group of . Since , it follows that is a behavior of . Thus, existence of behavior has been proven.
Let and be behaviors of . Then there exist generators and of the additive group of such that and . If , then , causing . If , then it must be the case that . (This follows from the fact that 1 and -1 are the only generators of .) Thus, , causing . Since and are nonnegative, it follows that . Thus, uniqueness of behavior has been proven. ∎
|Title||behavior exists uniquely (infinite case)|
|Date of creation||2013-03-22 16:02:32|
|Last modified on||2013-03-22 16:02:32|
|Last modified by||Wkbj79 (1863)|