# a finite ring is cyclic if and only its order and characteristic are equal

. A finite ring is cyclic if and only if its order (http://planetmath.org/OrderRing) and characteristic^{} are equal.

###### Proof.

If $R$ is a cyclic ring and $r$ is a generator^{} (http://planetmath.org/Generator) of the additive group^{} of $R$, then $|r|=|R|$. Since, for every $s\in R$, $|s|$ divides $|R|$, then it follows that $\mathrm{char}R=|R|$. Conversely, if $R$ is a finite ring such that $\mathrm{char}R=|R|$, then the exponent of the additive group of $R$ is also equal to $|R|$. Thus, there exists $t\in R$ such that $|t|=|R|$. Since $\u27e8t\u27e9$ is a subgroup^{} of the additive group of $R$ and $|\u27e8t\u27e9|=|t|=|R|$, it follows that $R$ is a cyclic ring.∎

Title | a finite ring is cyclic if and only its order and characteristic are equal |
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Canonical name | AFiniteRingIsCyclicIfAndOnlyItsOrderAndCharacteristicAreEqual |

Date of creation | 2013-03-22 13:30:30 |

Last modified on | 2013-03-22 13:30:30 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 12 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 13A99 |