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 $p$-adic topologies on the integers.

Definition
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 $$d(x,y) = 2^{-\sup{\{n \mid x-y\in\mathfrak{a}^nM\}}}.$$ The family $$\{x+\mathfrak{a}^nM:x\in M,n\in\mathbb{Z}^+\}$$ is a basis for this topology.

Properties
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$$\bigcap_{n > 0}{\mathfrak{a}^nM} = 0\text{,}$$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.

By Krull's intersection theorem, if $R$ is a Noetherian ring which is an integral domain or a local ring, it holds that $$\bigcap_{n > 0}{\mathfrak{a}^n} = 0$$ 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. This follows from the Artin-Rees lemma.

Completion
When $M$ is Hausdorff, $M$ can be completed as a metric space; the resulting space is denoted by $$\widehat M$$ and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to): $$\widehat{M} = \varprojlim M/\mathfrak{a}^n M$$ where the right-hand side is an inverse limit of quotient modules under natural projection.

For example, let $$R = k[x_1, \ldots, x_n]$$ be a polynomial ring over a field $k$ and $𝔞 = (x_{1}, ..., x_{n})$ the (unique) homogeneous maximal ideal. Then $$\hat{R} = kx_1, \ldots, x_n$$, the formal power series ring over $k$ in $n$ variables.

Closed submodules
As a consequence of the above, the $𝔞$-adic closure of a submodule $$N \subseteq M$$ is $\overline{N} = \bigcap_{n > 0}{(N + \mathfrak{a}^n M)}\text{.}$ This closure coincides with $N$ whenever $R$ is $𝔞$-adically complete and $M$ is finitely generated.

$R$ is called Zariski with respect to $𝔞$ if every ideal in $R$ is $𝔞$-adically closed. There is a characterization:
 * $R$ is Zariski with respect to $𝔞$ if and only if $𝔞$ is contained in the Jacobson radical of $R$.

In particular a Noetherian local ring is Zariski with respect to the maximal ideal.