# cyclic rings that are isomorphic to $k{\mathbb{Z}}_{kn}$

###### Corollary.

A finite cyclic ring of order (http://planetmath.org/OrderRing) $n$ with behavior $k$ is isomorphic^{} to $k\mathit{}{\mathrm{Z}}_{k\mathit{}n}$.

###### Proof.

Note that $k{\mathbb{Z}}_{kn}$ is a cyclic ring and that $k$ is a generator^{} of its additive group^{}. As groups, $k{\mathbb{Z}}_{kn}$ and ${\mathbb{Z}}_{n}$ are isomorphic. Thus, $k{\mathbb{Z}}_{kn}$ has order $n$. Since ${k}^{2}=k(k)$, then $k\mathbb{Z}$ has behavior $k$.
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Title | cyclic rings that are isomorphic to $k{\mathbb{Z}}_{kn}$ |
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Canonical name | CyclicRingsThatAreIsomorphicToKmathbbZkn |

Date of creation | 2013-03-22 16:02:45 |

Last modified on | 2013-03-22 16:02:45 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 10 |

Author | Wkbj79 (1863) |

Entry type | Corollary |

Classification | msc 16U99 |

Classification | msc 13M05 |

Classification | msc 13A99 |

Related topic | MathbbZ_n |