behavior exists uniquely (finite case)
Let be the order (http://planetmath.org/OrderRing) of and be a generator (http://planetmath.org/Generator) of the additive group of . Then there exists with . Let and with . Since , there exists with . Since , is 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 . Since is a generator of the additive group of , there exists with such that .
Note that . Thus, . Recall that . Therefore, . Since and are both positive divisors of and , it follows that . Thus, uniqueness of behavior has been proven. ∎
Note that it has also been shown that, if is a finite cyclic ring of order , is a generator of the additive group of , and with , then the behavior of is .
|Title||behavior exists uniquely (finite case)|
|Date of creation||2013-03-22 16:02:35|
|Last modified on||2013-03-22 16:02:35|
|Last modified by||Wkbj79 (1863)|