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.
|||O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975)|
|||N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)|
Adic topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adic_topology&oldid=16092