In mathematics, a Henselian ring (or Hensel ring) is a local ring

Local ring

In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...

 in which Hensel's lemma

Hensel's lemma

In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number , then this root corresponds to a unique root of the same equation modulo any higher power...

 holds. They were defined by , who named them after Kurt Hensel

Kurt Hensel

Kurt Wilhelm Sebastian Hensel was a German mathematician born in Königsberg, Prussia.He was the son of the landowner and entrepreneur Sebastian Hensel, brother of the philosopher Paul Hensel, grandson of the composer Fanny Mendelssohn and the painter Wilhelm Hensel, and a descendant of the...

.

Some standard references for Hensel rings are , , and .

Definitions

In this article Henselian rings will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings.

A commutative local ring R with maximal ideal

Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...

 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 ring is called strict, if its residue field

Residue field

In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field...

 is separably closed

Separable extension

In modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...

.

Henselian rings in algebraic geometry

Henselian rings are the local rings of "points" with respect to the Nisnevich topology

Nisnevich topology

In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives...

, 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 "points" of the étale topology

Étale topology

In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic...

.

Henselization

For any local ring A there is a universal Henselian ring B generated by A, called the Henselization of A, 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

Excellent ring

In mathematics, in the fields of commutative algebra and algebraic geometry, an excellent ring is a Noetherian commutative ring with many of the good properties of complete local rings...

then so is its Henselization.

Similarly there is a strictly Henselian ring generated by A, called the strict Henselization of A (which is not quite universal: it is unique, but only up to non-unique isomorphism).

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.

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.

Examples

• Any field is a Henselian local ring.
• Complete local rings, such as the ring of p-adic integers and rings of formal power series over a field, are Henselian.
• The rings of convergent power series over the real or complex numbers are Henselian.
• Rings of algebraic power series over a field are Henselian.
• A local ring that is integral over a Henselian ring is Henselian.
• The Henselization of a local ring is a Henselian local ring.
• Every quotient
  Quotient ring
  In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
  
   of a Henselian ring is Henselian.
• A ring A is Henselian if and only if the associated reduced ring
  Reduced ring
  In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0...
  
   A_red is Henselian (this is the quotient of A by the ideal of nilpotent elements).
• If A has only one prime ideal then it is Henselian since A_red is a field.