# criterion for cyclic rings to be principal ideal rings

###### Theorem.

A cyclic ring is a principal ideal ring if and only if it has a multiplicative identity^{}.

###### Proof.

Let $R$ be a cyclic ring. If $R$ has a multiplicative identity $u$, then $u$ generates (http://planetmath.org/Generator^{}) the additive group^{} of $R$. Let $I$ be an ideal of $R$. Since $\{{0}_{R}\}$ is principal, it may be assumed that $I$ contains a nonzero element. Let $n$ be the smallest natural number^{} such that $nu\in I$. The inclusion $\u27e8nu\u27e9\subseteq I$ is trivial. Let $t\in I$. Since $t\in R$, there exists $a\in \mathbb{Z}$ with $t=au$. By the division algorithm^{}, there exists $q,r\in \mathbb{Z}$ with $$ such that $a=qn+r$. Thus, $t=au=(qn+r)u=(qn)u+ru=q(nu)+ru$. Since $ru=t-q(nu)\in I$, by choice of $n$, it must be the case that $r=0$. Thus, $t=q(nu)$. Hence, $\u27e8nu\u27e9=I$, and $R$ is a principal ideal ring.

Conversely, if $R$ is a principal ideal ring, then $R$ is a principal ideal^{}. Let $k$ be the behavior of $R$ and $r$ be a generator (http://planetmath.org/Generator) of the additive group of $R$ such that ${r}^{2}=kr$. Since $R$ is principal, there exists $s\in R$ such that $\u27e8s\u27e9=R$. Let $a\in \mathbb{Z}$ such that $s=ar$. Since $r\in R=\u27e8s\u27e9$, there exists $t\in R$ with $st=r$. Let $b\in \mathbb{Z}$ such that $t=br$. Then $r=st=(ar)(br)=(ab){r}^{2}=(ab)(kr)=(abk)r$. If $R$ is infinite^{}, then $abk=1$, in which case $k=1$ since $k$ is nonnegative. If $R$ is finite, then $abk\equiv 1\mathrm{mod}|R|$, in which case $k=1$ since $k$ is a positive divisor^{} of $|R|$. In either case, $R$ has behavior one, and it follows that $R$ has a multiplicative identity.
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Title | criterion for cyclic rings to be principal ideal rings |

Canonical name | CriterionForCyclicRingsToBePrincipalIdealRings |

Date of creation | 2013-03-22 15:57:03 |

Last modified on | 2013-03-22 15:57:03 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 15 |

Author | Wkbj79 (1863) |

Entry type | Theorem |

Classification | msc 16U99 |

Classification | msc 13A99 |

Classification | msc 13F10 |

Related topic | CyclicRing3 |

Related topic | PrincipalIdealRing |

Related topic | MultiplicativeIdentityOfACyclicRingMustBeAGenerator |

Related topic | CyclicRingsOfBehaviorOne |