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$I$-adic topology (Definition)

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

$\displaystyle \bigcap_{k=1}^\infty I^k=\{0\}.$    

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$,

$\displaystyle d_I(r_1,r_2)=2^{-ord_I(r_1-r_s)}.$    

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\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$.



"$I$-adic topology" is owned by mathcam.
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Other names:  I-adic topology
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Cross-references: converges, Cauchy sequence, complete, parallel, sequence, equivalent, induce, real number, number, induced, topology, entire, intersection, well-defined, integer, metric, ideal, ring

This is version 4 of $I$-adic topology, born on 2004-09-18, modified 2004-09-21.
Object id is 6193, canonical name is IAdicTopology.
Accessed 2113 times total.

Classification:
AMS MSC: 13B35 (Commutative rings and algebras :: Ring extensions and related topics :: Completion)
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Viewing Problem by mathcam on 2004-09-21 10:30:11
I got this mail:

> Your entry, at less in my browser, appears something like that:
> ``x(enclosed in a square)$I$-adic topology''. How do I fix it? Thanks > for your help.

Anyone have any thoughts?

Thanks,

Cam
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