Editing I-adic topology (section)

==Properties==
With respect to the topology, the module operations of addition and scalar multiplication are [[Continuity (topology)|continuous]], so that {{mvar|M}} becomes a [[topological module]].  However, {{mvar|M}} need not be [[Hausdorff space|Hausdorff]]; it is Hausdorff [[if and only if]]<math display=block>\bigcap_{n > 0}{\mathfrak{a}^nM} = 0\text{,}</math>so that {{mvar|d}} becomes a genuine [[metric (mathematics)|metric]]. Related to the usual terminology in topology, where a Hausdorff space is also called separated, in that case, the {{mvar|𝔞}}-adic topology is called ''separated''.{{sfn|Singh|2011|p=147}}

By [[Krull's intersection theorem]], if {{mvar|R}} is a [[Noetherian ring]] which is an [[integral domain]] or a [[local ring]], it holds that <math>\bigcap_{n > 0}{\mathfrak{a}^n} = 0</math> for any proper ideal {{mvar|𝔞}} of {{mvar|R}}. Thus under these conditions, for any proper ideal {{mvar|𝔞}} of {{mvar|R}} and any {{mvar|R}}-module {{mvar|M}}, the {{mvar|𝔞}}-adic topology on {{mvar|M}} is separated.   

For a submodule {{mvar|N}} of {{mvar|M}}, the [[1st isomorphism theorem|canonical]] [[homomorphism]] to {{math|''M''/''N''}} induces a [[quotient topology]] which coincides with the {{math|𝔞}}-adic topology. The analogous result is not necessarily true for the submodule {{mvar|N}} itself: the [[subspace topology]] need not be the {{math|𝔞}}-adic topology. However, the two topologies coincide when {{mvar|R}} is Noetherian and {{mvar|M}} [[finitely generated module|finitely generated]]. This follows from the [[Artin–Rees lemma]].{{sfn|Singh|2011|p=148}}