Editing Commutative ring (section)

== Constructing commutative rings ==
There are several ways to construct new rings out of given ones. The aim of such constructions is often to improve certain properties of the ring so as to make it more readily understandable. For example, an integral domain that is [[integral element#Equivalent definitions|integrally closed]] in its [[field of fractions]] is called [[normal ring|normal]]. This is a desirable property, for example any normal one-dimensional ring is necessarily [[Regular local ring|regular]]. Rendering{{clarify|date=March 2012}} a ring normal is known as ''normalization''.

=== Completions ===
If ''I'' is an ideal in a commutative ring ''R'', the powers of ''I'' form [[neighborhood (topology)|topological neighborhoods]] of ''0'' which allow ''R'' to be viewed as a [[topological ring]]. This topology is called the [[I-adic topology|''I''-adic topology]]. ''R'' can then be completed with respect to this topology. Formally, the ''I''-adic completion is the [[inverse limit]] of the rings ''R''/''I<sup>n</sup>''. For example, if ''k'' is a field, ''k''<nowiki>[[</nowiki>''X''<nowiki>]]</nowiki>, the [[formal power series]] ring in one variable over ''k'', is the ''I''-adic completion of ''k''[''X''] where ''I'' is the principal ideal generated by ''X''. This ring serves as an algebraic analogue of the disk. Analogously, the ring of [[p-adic number|''p''-adic integers]] is the completion of '''Z'''  with respect to the principal ideal (''p''). Any ring that is isomorphic to its own completion, is called [[complete ring|complete]].

Complete local rings satisfy [[Hensel's lemma]], which roughly speaking allows extending solutions (of various problems) over the residue field ''k'' to ''R''.