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A 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), 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

 [1] O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) [2] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
How to Cite This Entry:
Adic topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adic_topology&oldid=45034
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article