Editing Local ring (section)

=== Commutative case===

We also write {{nowrap|(''R'', ''m'')}} for a commutative local ring ''R'' with maximal ideal ''m''. Every such ring becomes a [[topological ring]] in a natural way if one takes the powers of ''m'' as a [[neighborhood base]] of 0. This is the [[I-adic topology|''m''-adic topology]] on ''R''. If {{nowrap|(''R'', ''m'')}} is a commutative [[Noetherian ring|Noetherian]] local ring, then

:<math>\bigcap_{i=1}^\infty m^i = \{0\}</math>
('''Krull's intersection theorem'''), and it follows that ''R'' with the ''m''-adic topology is a [[Hausdorff space]]. The theorem is a consequence of the [[Artin–Rees lemma]] together with [[Nakayama's lemma]], and, as such, the "Noetherian" assumption is crucial. Indeed, let ''R'' be the ring of germs of infinitely differentiable functions at 0 in the real line and ''m'' be the maximal ideal <math>(x)</math>. Then a nonzero function <math>e^{-{1 \over x^2}}</math> belongs to <math>m^n</math> for any ''n'', since that function divided by <math>x^n</math> is still smooth.

As for any topological ring, one can ask whether {{nowrap|(''R'', ''m'')}} is [[Complete uniform space|complete]] (as a [[uniform space]]); if it is not, one considers its [[Completion (ring theory)|completion]], again a local ring. Complete Noetherian local rings are classified by the [[Cohen structure theorem]].

In [[algebraic geometry]], especially when ''R'' is the local ring of a scheme at some point ''P'', {{nowrap|''R'' / ''m''}} is called the ''[[residue field]]'' of the local ring or residue field of the point ''P''.

If {{nowrap|(''R'', ''m'')}} and {{nowrap|(''S'', ''n'')}} are local rings, then a '''local ring homomorphism''' from ''R'' to ''S'' is a [[ring homomorphism]] {{nowrap|''f'' : ''R'' → ''S''}} with the property {{nowrap|''f''(''m'') ⊆ ''n''}}.<ref>{{Cite web|url=http://stacks.math.columbia.edu/tag/07BI|title=Tag 07BI}}</ref> These are precisely the ring homomorphisms that are continuous with respect to the given topologies on ''R'' and ''S''. For example, consider the ring morphism <math>\mathbb{C}[x]/(x^3) \to \mathbb{C}[x,y]/(x^3,x^2y,y^4)</math> sending <math>x \mapsto x</math>. The preimage of <math>(x,y)</math> is <math>(x)</math>. Another example of a local ring morphism is given by <math>\mathbb{C}[x]/(x^3) \to \mathbb{C}[x]/(x^2)</math>.