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 𝔞 of R determines a topology on M called the 𝔞-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.
so that d becomes a genuine metric.
Related to the usual terminology in topology, where a Hausdorff space is also called separated, in that case, the 𝔞-adic topology is called separated.
[1] 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 𝔞 of R. Thus under these conditions, for any proper ideal 𝔞 of R and any R-module M, the 𝔞-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 𝔞-adic topology.
The analogous result is not necessarily true for the submodule N itself: the subspace topology need not be the 𝔞-adic topology.
However, the two topologies coincide when R is Noetherian and M finitely generated.
[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.
where the right-hand side is an inverse limit of quotient modules under natural projection.
be a polynomial ring over a field k and 𝔞 = (x1, ..., xn) the (unique) homogeneous maximal ideal.
, the formal power series ring over k in n variables.
[4] The 𝔞-adic closure of a submodule
[5] This closure coincides with N whenever R is 𝔞-adically complete and M is finitely generated.
[6] R is called Zariski with respect to 𝔞 if every ideal in R is 𝔞-adically closed.
There is a characterization: In particular a Noetherian local ring is Zariski with respect to the maximal ideal.