# $I$-adic topology

Let $R$ be a ring and $I$ an ideal in $R$ such that

$$\bigcap _{k=1}^{\mathrm{\infty}}{I}^{k}=\{0\}.$$ |

Though not usually explicitly done, we can define a metric on $R$ by defining $or{d}_{I}(r)$ for a $r\in R$ by $or{d}_{I}(r)=k$ where $k$ is the largest integer such that $r\in {I}^{k}$ (well-defined by the intersection^{} assumption^{}, and ${I}^{0}$ is taken to be the entire ring) and by $or{d}_{I}(0)=\mathrm{\infty}$, and then defining for any ${r}_{1},{r}_{2}\in R$,

$${d}_{I}({r}_{1},{r}_{2})={2}^{-or{d}_{I}({r}_{1}-{r}_{s})}.$$ |

The topology^{} induced by this metric is called the $I$-adic topology. Note that the number 2 was chosen rather arbitrarily. Any other real number greater than 1 will induce an equivalent^{} topology.

Except in the case of the similarly-defined $p$-adic topology, it is rare that reference is made to the actual $I$-adic metric. Instead, we usually refer to the $I$-adic topology.

In particular, a sequence of elements in $\{{r}_{i}\}\in R$ is Cauchy with respect to this topology if for any $k$ there exists an $N$ such that for all $m,n\ge N$ we have $({a}_{m}-{a}_{n})\in {I}^{k}$. (Note the parallel with the metric version of Cauchy, where $k$ plays the part analogous to an arbitrary $\u03f5$). The ring $R$ is complete^{} with respect to the $I$-adic topology if every such Cauchy sequence^{} converges^{} to an element of $R$.

Title | $I$-adic topology |
---|---|

Canonical name | IadicTopology |

Date of creation | 2013-03-22 14:36:59 |

Last modified on | 2013-03-22 14:36:59 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 13B35 |

Synonym | I-adic topology |