# maximal ideals of ring of formal power series

Suppose that $R$ is a commutative ring with non-zero unity.

If $\mathrm{\pi \x9d\x94\u037a}$ is a maximal ideal^{} of $R$, thenβ $\mathrm{\pi \x9d\x94\x90}:=\mathrm{\pi \x9d\x94\u037a}+(X)$β is a maximal ideal of the ring $R\beta \x81\u2019[[X]]$ of formal power series.

Also the converse is true, i.e. if $\mathrm{\pi \x9d\x94\x90}$ is a maximal ideal of $R\beta \x81\u2019[[X]]$, then there is a maximal ideal
$\mathrm{\pi \x9d\x94\u037a}$ of $R$ such thatβ $\mathrm{\pi \x9d\x94\x90}=\mathrm{\pi \x9d\x94\u037a}+(X)$.

Note.β In the special case that $R$ is a field, the only maximal ideal of which is the zero ideal^{} $(0)$, this corresponds to the only maximal ideal $(X)$ of $R\beta \x81\u2019[[X]]$ (see http://planetmath.org/node/12087formal power series over field).

We here prove the first assertion.β So, $\mathrm{\pi \x9d\x94\u037a}$ is assumed to be maximal.β Let

$$f\beta \x81\u2019(x):={a}_{0}+{a}_{1}\beta \x81\u2019X+{a}_{2}\beta \x81\u2019{X}^{2}+\mathrm{\beta \x80\xa6}$$ |

be any formal power series in $R\beta \x81\u2019[[X]]\beta \x88\x96\mathrm{\pi \x9d\x94\x90}$.β Hence, the constant term ${a}_{0}$ cannot lie in $\mathrm{\pi \x9d\x94\u037a}$.β According to the criterion for maximal ideal, there is an element $r$ of $R$ such thatβ $1+r\beta \x81\u2019{a}_{0}\beta \x88\x88\mathrm{\pi \x9d\x94\u037a}$.β Therefore

$$1+r\beta \x81\u2019f\beta \x81\u2019(X)=(1+r\beta \x81\u2019{a}_{0})+r\beta \x81\u2019({a}_{1}+{a}_{2}\beta \x81\u2019X+{a}_{3}\beta \x81\u2019{X}^{2}+\mathrm{\beta \x80\xa6})\beta \x81\u2019X\beta \x88\x88\mathrm{\pi \x9d\x94\u037a}+(X)=\mathrm{\pi \x9d\x94\x90},$$ |

whence the same criterion says that $\mathrm{\pi \x9d\x94\x90}$ is a maximal ideal of $R\beta \x81\u2019[[X]]$.

Title | maximal ideals of ring of formal power series |
---|---|

Canonical name | MaximalIdealsOfRingOfFormalPowerSeries |

Date of creation | 2013-03-22 19:10:49 |

Last modified on | 2013-03-22 19:10:49 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Result |

Classification | msc 13H05 |

Classification | msc 13J05 |

Classification | msc 13C13 |

Classification | msc 13F25 |