View source for Adic topology

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A [[Linear topology|linear topology]] of a ring  $  A $
in which the fundamental system of neighbourhoods of zero consists of the powers  $  \mathfrak A  ^ {n} $
of some two-sided ideal  $  \mathfrak A $. 
The topology is then said to be  $  \mathfrak A $-
adic, and the ideal  $  \mathfrak A $
is said to be the defining ideal of the topology. The closure of any set  $  F \subset  A $
in the  $  \mathfrak A $-
adic topology is equal to  $  \cap _ {n \geq  0 }  ( F + \mathfrak A  ^ {n} ) $; 
in particular, the topology is separable if, and only if,  $  \cap _ {n \geq  0 }  \mathfrak A  ^ {n} = (0) $. 
The separable completion  $  \widehat{A}  $
of the ring  $  A $
in an  $  \mathfrak A $-
adic topology is isomorphic to the projective limit  $  \lim\limits _  \leftarrow  ( A / \mathfrak A  ^ {n} ) $.

The  $  \mathfrak A $-
adic topology of an  $  A $-
module  $  M $
is defined in a similar manner: its fundamental system of neighbourhoods of zero is given by the submodules  $  \mathfrak A  ^ {n} M $; 
in the  $  \mathfrak A $-
adic topology  $  M $
becomes a topological  $  A $-
module.

Let  $  A $
be a commutative ring with identity with an  $  \mathfrak A $-
adic topology and let  $  \widehat{A}  $
be its completion; if  $  \mathfrak A $
is an ideal of finite type, the topology in  $  \widehat{A}  $
is  $  \widehat{\mathfrak A}  $-
adic, and  $  {\widehat{\mathfrak A}  } {}  ^ {n} = \mathfrak A  ^ {n} \widehat{A}  $. 
If  $  \mathfrak A $
is a maximal ideal, then  $  \widehat{A}  $
is a local ring with maximal ideal  $  \widehat{\mathfrak A}  $. 
A local ring topology is an adic topology defined by its maximal ideal (an  $  \mathfrak m $-
adic topology).

A fundamental tool in the study of adic topologies of rings is the Artin–Rees lemma: Let  $  A $
be a commutative Noetherian ring, let  $  \mathfrak A $
be an ideal in  $  A $, 
let  $  E $
be an  $  A $-
module of finite type, and let  $  F $
be a submodule of  $  E $. 
Then there exists a  $  k $
such that, for any  $  n \geq  0 $, 
the following equality is valid:

$$ 
\mathfrak A  ^ {n} ( \mathfrak A  ^ {k} E \cap F )  = \ 
\mathfrak A ^ {k + n } E \cap F .
$$

The topological interpretation of the Artin–Rees lemma shows that the  $  \mathfrak A $-
adic topology of  $  F $
is induced by the  $  \mathfrak A $-
adic topology of  $  E $. 
It follows that the completion  $  \widehat{A}  $
of a ring  $  A $
in the  $  \mathfrak A $-
adic topology is a flat  $  A $-
module (cf. [[Flat module|Flat module]]), that the completion  $  \widehat{E}  $
of the  $  A $-
module  $  E $
of finite type is identical with  $  E \otimes _ {A} \widehat{A}  $, 
and that Krull's theorem holds: The  $  \mathfrak A $-
adic topology of a Noetherian ring is separable if and only if the set  $  1 + \mathfrak A $
contains no zero divisors. In particular, the topology is separable if  $  \mathfrak A $
is contained in the (Jacobson) radical of the ring.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,   P. Samuel,   "Commutative algebra" , '''2''' , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>