Adic topology

A linear topology of a ring in which the fundamental system of neighbourhoods of zero consists of the powers of some two-sided ideal . The topology is then said to be -adic, and the ideal is said to be the defining ideal of the topology. The closure of any set in the -adic topology is equal to ; in particular, the topology is separable if, and only if, . The separable completion of the ring in an -adic topology is isomorphic to the projective limit .

The -adic topology of an -module is defined in a similar manner: its fundamental system of neighbourhoods of zero is given by the submodules ; in the -adic topology becomes a topological -module.

Let be a commutative ring with identity with an -adic topology and let be its completion; if is an ideal of finite type, the topology in is -adic, and . If is a maximal ideal, then is a local ring with maximal ideal . A local ring topology is an adic topology defined by its maximal ideal (an -adic topology).

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

The topological interpretation of the Artin–Rees lemma shows that the -adic topology of is induced by the -adic topology of . It follows that the completion of a ring in the -adic topology is a flat -module (cf. Flat module), that the completion of the -module of finite type is identical with , and that Krull's theorem holds: The -adic topology of a Noetherian ring is separable if and only if the set contains no zero divisors. In particular, the topology is separable if 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)