Completion (ring theory)

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In commutative algebra, the term completion refers to several related functors on topological rings and modules.

[edit] I-adic topology

The most important case is the so-called I-adic topology on a commutative ring R associated to an ideal I.

In it, a basis of open neighbourhoods of 0 is given by the powers

Iⁿ,

which are nested (they are smaller as n gets larger, because I is an ideal). Speaking more formally they form a descending filtration by :

$F^0{R}=R\supset I\supset I^2\supset\ldots, \quad F^n{R}=I^n.$

The completion $\hat{R}$ (pronounced "R hat") of the ring R is the inverse limit of the rings

R/IⁿR

as n goes to infinity. As may be expected, this produces a complete topological ring.

There is a related topology also on R-modules (also called I-adic): a basis of open neighborhoods of a module M is given by the sets of the form

x + IⁿM

for x ∈ M. The completion of an R-module M is an $\hat{R}$ -module $\hat{M}$ obtained as the inverse limit of the quotients

M/IⁿM.

This converts any module over R into a complete topological module over $\hat{R}$ .

[edit] Examples

1. The ring of p-adic integers Z_p is obtained by completing the ring Z of integers at the ideal (p).

2. Let R be the ring of polynomials in n variables over a field K and I be the maximal ideal generated by the variables. Then the completion of R at the ideal I is the ring of the formal power series over K.