# behavior

If $R$ is an infinite cyclic ring (http://planetmath.org/CyclicRing3), the behavior of $R$ is a nonnegative integer $k$ such that there exists a generator^{} (http://planetmath.org/Generator) $r$ of the additive group^{} of $R$ with ${r}^{2}=kr$.

If $R$ is a finite cyclic ring of order $n$, the behavior of $R$ is a positive divisor^{} $k$ of $n$ such that there exists a generator $r$ of the additive group of $R$ with ${r}^{2}=kr$.

For any cyclic ring, behavior exists uniquely. Moreover, the behavior of a cyclic ring determines many of its .

To the best of my knowledge, this definition first appeared in my master’s thesis:

Buck, Warren. *http://planetmath.org/?op=getobj&from=papers&id=336Cyclic Rings*. Charleston, IL: Eastern Illinois University, 2004.

Title | behavior |
---|---|

Canonical name | Behavior |

Date of creation | 2013-03-22 16:02:29 |

Last modified on | 2013-03-22 16:02:29 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 15 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 13A99 |

Classification | msc 16U99 |