Editing Ring (mathematics) (section)

=== Completion ===
Let {{mvar|R}} be a commutative ring, and let {{mvar|I}} be an ideal of&nbsp;{{mvar|R}}.
The '''[[Completion (ring theory)|completion]]''' of {{mvar|R}} at {{mvar|I}} is the projective limit <math>\hat{R} = \varprojlim R/I^n;</math> it is a commutative ring. The canonical homomorphisms from {{mvar|R}} to the quotients <math>R/I^n</math> induce a homomorphism <math>R \to \hat{R}.</math> The latter homomorphism is injective if {{mvar|R}} is a Noetherian integral domain and {{mvar|I}} is a proper ideal, or if {{mvar|R}} is a Noetherian local ring with maximal ideal {{mvar|I}}, by [[Krull's intersection theorem]].{{sfnp|Atiyah|Macdonald|1969|loc=Theorem 10.17 and its corollaries|ps=}} The construction is especially useful when {{mvar|I}} is a maximal ideal.

The basic example is the completion of {{tmath|\Z}} at the principal ideal {{math|(''p'')}} generated by a prime number {{mvar|p}}; it is called the ring of [[p-adic integer|{{mvar|p}}-adic integers]] and is denoted {{tmath|\Z_p.}} The completion can in this case be constructed also from the [[p-adic absolute value|{{mvar|p}}-adic absolute value]] on {{tmath|\Q.}} The {{mvar|p}}-adic absolute value on {{tmath|\Q}} is a map <math>x \mapsto |x|</math> from {{tmath|\Q}} to {{tmath|\R}} given by <math>|n|_p=p^{-v_p(n)}</math> where <math>v_p(n)</math> denotes the exponent of {{mvar|p}} in the prime factorization of a nonzero integer {{mvar|n}} into prime numbers (we also put <math>|0|_p=0</math> and <math>|m/n|_p = |m|_p/|n|_p</math>). It defines a distance function on {{tmath|\Q}} and the completion of {{tmath|\Q}} as a [[metric space]] is denoted by {{tmath|\Q_p.}} It is again a field since the field operations extend to the completion. The subring of {{tmath|\Q_p}} consisting of elements {{mvar|x}} with {{math|{{abs|''x''}}{{sub|''p''}} ≤ 1}} is isomorphic to&nbsp;{{tmath|\Z_p.}}

Similarly, the formal power series ring {{math|''R''[{[''t'']}]}} is the completion of {{math|''R''[''t'']}} at {{math|(''t'')}} (see also ''[[Hensel's lemma]]'')<!-- need to be discussed with concrete examples. Start of a para: The notion of the completion formalizes the construction of a solution by successive approximation. -->

A complete ring has much simpler structure than a commutative ring. This owns to the [[Cohen structure theorem]], which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the [[integral closure]] and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of [[excellent ring]].