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 p-adic topologies on the integers.
Let R be a commutative ring and M an R-module. Then each ideal ๐ of R determines a topology on M called the ๐-adic topology, characterized by the pseudometric
With respect to the topology, the module operations of addition and scalar multiplication are continuous, so that M becomes a topological module. However, M need not be Hausdorff; it is Hausdorff if and only if
By Krull's intersection theorem, if R is a Noetherian ring which is an integral domain or a local ring, it holds that for any proper ideal ๐ of R. Thus under these conditions, for any proper ideal ๐ of R and any R-module M, the ๐-adic topology on M is separated.
For a submodule N of M, the canonical homomorphism to M/N induces a quotient topology which coincides with the ๐-adic topology. The analogous result is not necessarily true for the submodule N itself: the subspace topology need not be the ๐-adic topology. However, the two topologies coincide when R is Noetherian and M finitely generated. This follows from the Artin-Rees lemma. [2]
When M is Hausdorff, M 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):
For example, let be a polynomial ring over a field k and ๐ = (x1, ..., xn) the (unique) homogeneous maximal ideal. Then , the formal power series ring over k in n variables. [4]
As a consequence of the above, the ๐-adic closure of a submodule is [5] This closure coincides with N whenever R is ๐-adically complete and M is finitely generated. [6]
R is called Zariski with respect to ๐ if every ideal in R is ๐-adically closed. There is a characterization:
In particular a Noetherian local ring is Zariski with respect to the maximal ideal. [7]