# cyclic rings of behavior one

###### Theorem.

A cyclic ring has a multiplicative identity^{} if and only if it has behavior one.

###### Proof.

For a proof that a cyclic ring with a multiplicative identity has behavior one, see this theorem^{} (http://planetmath.org/MultiplicativeIdentityOfACyclicRingMustBeAGenerator).

Let $R$ be a cyclic ring with behavior one. Let $r$ be a generator^{} (http://planetmath.org/Generator) of the additive group^{} of $R$ such that ${r}^{2}=r$. Let $s\in R$. Then there exists $a\in R$ with $s=ar$. Since $rs=r(ar)=a{r}^{2}=ar=s$ and multiplication^{} in cyclic rings is commutative^{}, then $r$ is a multiplicative identity.
∎

Title | cyclic rings of behavior one |
---|---|

Canonical name | CyclicRingsOfBehaviorOne |

Date of creation | 2013-03-22 16:03:10 |

Last modified on | 2013-03-22 16:03:10 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 9 |

Author | Wkbj79 (1863) |

Entry type | Theorem |

Classification | msc 13A99 |

Classification | msc 16U99 |

Classification | msc 13F10 |

Related topic | MultiplicativeIdentityOfACyclicRingMustBeAGenerator |

Related topic | CriterionForCyclicRingsToBePrincipalIdealRings |