criteria for cyclic rings to be isomorphic
If and have the same order and the same behavior, then let be a generator of the additive group of with . Define by for every . This map is clearly well defined and surjective. Since and have the same order, is injective. Since, for every , and
it follows that is an isomorphism.
Conversely, let be an isomorphism. Then and must have the same order. If is infinite, then is infinite, and is a nonnegative integer. If is finite, then divides (http://planetmath.org/Divisibility) , which equals . In either case, is a candidate for the behavior of . Since is a generator of the additive group of and is an isomorphism, is a generator of the additive group of . Since , it follows that has behavior . ∎
|Title||criteria for cyclic rings to be isomorphic|
|Date of creation||2013-03-22 16:02:39|
|Last modified on||2013-03-22 16:02:39|
|Last modified by||Wkbj79 (1863)|