I-adic topology
In commutative algebra, the mathematical study of commutative rings, adic topologies are a family of topologies on the underlying set of a module, generalizing the -adic topologies on the integers.
Definition
Let be a commutative ring and an -module. Then each ideal of determines a topology on called the -adic topology, characterized by the pseudometric The family is a basis for this topology.An -adic topology is a linear topology.
Properties
With respect to the topology, the module operations of addition and scalar multiplication are continuous, so that becomes a topological module. However, need not be Hausdorff; it is Hausdorff if and only ifso that becomes a genuine metric. Related to the usual terminology in topology, where a Hausdorff space is also called separated, in that case, the -adic topology is called separated.By Krull's intersection theorem, if is a Noetherian ring which is an integral domain or a local ring, it holds that for any proper ideal of. Thus under these conditions, for any proper ideal of and any -module, the -adic topology on is separated.
For a submodule of, the canonical homomorphism to induces a quotient topology which coincides with the -adic topology. The analogous result is not necessarily true for the submodule itself: the subspace topology need not be the -adic topology. However, the two topologies coincide when is Noetherian and finitely generated. This follows from the Artin–Rees lemma.
Completion
When is Hausdorff, can be [completion of a metric (mathematics)|metric space|completed] as a metric space; the resulting space is denoted by and has the module structure obtained by extending the module operations by continuity. It is also the same as : where the right-hand side is an inverse limit of quotient modules under natural projection.For example, let be a polynomial ring over a field and the homogeneous maximal ideal. Then, the formal power series ring over in variables.
Closed submodules
The -adic closure of a submodule is This closure coincides with whenever is -adically complete and is finitely generated.is called Zariski with respect to if every ideal in is -adically closed. There is a characterization:
In particular a Noetherian local ring is Zariski with respect to the maximal ideal.