IAdicTopology 2004-09-18 12:29:20 2004-09-21 16:49:18 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} Let $R$ be a ring and $I$ an ideal in $R$ such that \begin{equation*} \bigcap_{k=1}^\infty I^k=\{0\}. \end{equation*} Though not usually explicitly done, we can define a metric on $R$ by defining $ord_I(r)$ for a $r\in R$ by $ord_I(r)=k$ 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 $ord_I(0)=\infty$, and then defining for any $r_1,r_2\in R$, \begin{equation*} d_I(r_1,r_2)=2^{-ord_I(r_1-r_s)}. \end{equation*} 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 $m,n\geq N$ 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 $\epsilon$). The ring $R$ is complete with respect to the $I$-adic topology if every such Cauchy sequence converges to an element of $R$.