Henselian local ring

This article defines a property that can be evaluated for a local ring
View other properties of local rings

Definition

Symbol-free definition

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:

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

we can then find a $b$ such that:

$f(b)=0$ and $b\equiv a(mod{f}^{\prime }(a)I)$

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