Henselian local ring: Difference between revisions

Revision as of 12:14, 7 August 2007

This article defines a property that can be evaluated for a local ring
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Let $R$ be a local ring with maximal ideal $I$ . We say that $R$ is Henselian if given any polynomial $f(x)\in R[x]$ and $a$ such that:

$f(a)\equiv 0(\mod f'(a)^{2}I)$

we can then find a $b$ such that:

$f(b)=0$ and $b\equiv a(\mod f'(a)I)$

Another way of putting it is that the ring must satisfy Hensel's lemma with respect to its unique maximal ideal.

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 Let <math>R</math> be a [[local ring]] with maximal ideal <math>I</math>. We say that <math>R</math> is ''Henselian'' if given any polynomial <math>f(x) \in R[x]</math> and <math>a</math> such that:
-<math>f(a) \equiv 0 (mod f'(a)^2I)</math>
+<math>f(a) \equiv 0 (\mod f'(a)^2I)</math>
 we can then find a <math>b</math> such that:
-<math>f(b) = 0</math> and <math>b \equiv a (mod f'(a)I)</math>
+<math>f(b) = 0</math> and <math>b \equiv a (\mod f'(a)I)</math>
 Another way of putting it is that the ring must satisfy [[Hensel's lemma]] with respect to its unique maximal ideal.