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.

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 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 (or canonically isomorphic to):  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 (unique) homogeneous maximal ideal. Then , the formal power series ring over  in  variables.

Closed submodules
As a consequence of the above, 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:
 is Zariski with respect to  if and only if  is contained in the Jacobson radical of .
In particular a Noetherian local ring is Zariski with respect to the maximal ideal.

References

Sources
 
 

Commutative algebra
Topology