# regular ideal

An ideal $\U0001d51e$ of a ring $R$ is called a , iff $\U0001d51e$ a regular element^{} of $R$.

Proposition. If $m$ is a positive integer, then the only regular ideal in the residue class ring ${\mathbb{Z}}_{m}$ is the unit ideal $(1)$.

Proof. The ring ${\mathbb{Z}}_{m}$ is a principal ideal ring. Let $(n)$ be any regular ideal of the ring ${\mathbb{Z}}_{m}$. Then $n$ can not be zero divisor, since otherwise there would be a non-zero element $r$ of ${\mathbb{Z}}_{m}$ such that $nr=0$ and thus every element $sn$ of the principal ideal^{} would satisfy $(sn)r=s(nr)=s0=0$. So, $n$ is a regular element of ${\mathbb{Z}}_{m}$ and therefore we have $\mathrm{gcd}(m,n)=1$. Then, according to Bézout’s lemma (http://planetmath.org/BezoutsLemma), there are such integers $x$ and $y$ that $1=xm+yn$. This equation gives the congruence^{} $1\equiv yn\phantom{\rule{veryverythickmathspace}{0ex}}(modm)$, i.e. $1=yn$ in the ring ${\mathbb{Z}}_{m}$. With $1$ the principal ideal $(n)$ contains all elements of ${\mathbb{Z}}_{m}$, which means that $(n)={\mathbb{Z}}_{m}=(1)$.

Note. The above notion of “regular ideal” is used in most books concerning ideals of commutative rings, e.g. [1]. There is also a different notion of “regular ideal” mentioned in [2] (p. 179): Let $I$ be an ideal of the commutative ring $R$ with non-zero unity. This ideal is called regular, if the quotient ring^{} $R/I$ is a regular ring^{}, in other words, if for each $a\in R$ there exists an element $b\in R$ such that
${a}^{2}b-a\in I$.

## References

- 1 M. Larsen and P. McCarthy: “Multiplicative theory of ideals”. Academic Press. New York (1971).
- 2 D. M. Burton: “A first course in rings and ideals”. Addison-Wesley. Reading, Massachusetts (1970).

Title | regular ideal |
---|---|

Canonical name | RegularIdeal |

Date of creation | 2013-03-22 15:43:05 |

Last modified on | 2013-03-22 15:43:05 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 12 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 14K99 |

Classification | msc 16D25 |

Classification | msc 11N80 |

Classification | msc 13A15 |

Related topic | QuasiRegularIdeal |

Related topic | QuasiRegularity |