Henselian ring

In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now restrict them to be commutative.
Some standard references for Hensel rings are , , and .
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"["x"], then any factorization of its image "P" in ("R"/"m")["x"] into a product of coprime monic polynomials can be lifted to a factorization in "R"["x"].
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.
Henselian rings in algebraic geometry.
Henselian rings are the local rings of "points" with respect to the Nisnevich topology, so the spectra of these rings do not admit non-trivial connected coverings with respect to the Nisnevich topology. Likewise strict Henselian rings are the local rings of geometric points in the étale topology.
Henselization.
For any local ring "A" there is a universal Henselian ring "B" generated by "A", called the Henselization of "A", introduced by , such that any local homomorphism from "A" to a Henselian ring can be extended uniquely to "B". The Henselization of "A" is unique up to unique isomorphism. The Henselization of "A" is an algebraic substitute for the completion of "A". The Henselization of "A" has the same completion and residue field as "A" and is a flat module over "A". If "A" is Noetherian, reduced, normal, regular, or excellent then so is its Henselization. For example, the Henselization of the ring of polynomials "k"["x","y"...] localized at the point (0,0...) is the ring of algebraic formal power series (the formal power series satisfying an algebraic equation). This can be thought of as the "algebraic" part of the completion.
Similarly there is a strictly Henselian ring generated by "A", called the strict Henselization of "A". The strict Henselization is not quite universal: it is unique, but only up to non-unique isomorphism. More precisely it depends on the choice of a separable algebraic closure of the residue field of "A", and automorphisms of this separable algebraic closure correspond to automorphisms of the corresponding strict Henselization. For example, a strict Henselization of the field of "p"-adic numbers is given by the maximal unramified extension, generated by all roots of unity of order prime to "p". It is not "universal" as it has non-trivial automorphisms.