# proof that every subring of a cyclic ring is a cyclic ring

The following is a proof that every subring of a cyclic ring is a cyclic ring.

###### Proof.

Let $R$ be a cyclic ring and $S$ be a subring of $R$. Then the additive group^{} of $S$ is a subgroup^{} of the additive group of $R$. By definition of cyclic group^{}, the additive group of $R$ is cyclic (http://planetmath.org/CyclicGroup). Thus, the additive group of $S$ is cyclic. It follows that $S$ is a cyclic ring.
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Title | proof that every subring of a cyclic ring is a cyclic ring |
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Canonical name | ProofThatEverySubringOfACyclicRingIsACyclicRing |

Date of creation | 2013-03-22 13:30:50 |

Last modified on | 2013-03-22 13:30:50 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 8 |

Author | Wkbj79 (1863) |

Entry type | Proof |

Classification | msc 13A99 |

Classification | msc 16U99 |

Related topic | ProofThatAllSubgroupsOfACyclicGroupAreCyclic |

Related topic | ProofThatEverySubringOfACyclicRingIsAnIdeal |