Definitions
In this article rings will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings.
A local ring R with maximal ideal m is called Henselian if Hensel's lemma holds. This means that if P is a monic polynomial in R, then any factorization of its image P in (R/m) into a product of coprime monic polynomials can be lifted to a factorization in R.
A local ring is Henselian if and only if every finite ring extension is a product of local rings.
A Henselian local ring is called strictly Henselian if its residue field is separably closed.
A field with valuation is said to be Henselian if its valuation ring is Henselian.
A ring is called Henselian if it is a direct product of a finite number of Henselian local rings.
Read more about this topic: Henselian Ring
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