Henselian ring

In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were defined by harvtxt|Azumaya|1951, who named them after Kurt Hensel.

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 "m" is called Henselian if Hensel's lemma holds. This means that if "P" is a monic polynomial, then any factorization of "P" in "R"/"m" into a product of coprime monic polynomials can be lifted to 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 is separably closed.

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 "points" of the étale topology.

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 excellentthen so is its Henselization.

Similarly there is a strictly Henselian ring generated by "A", called the strict Henselization of "A" (which 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 atthe 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 ring of "p"-adic integers 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 of a Henselian ring is Henselian.
*A ring "A" is Henselian if and only if the associated reduced ring "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.

References

Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

Weierstrass ring — In mathematics, a Weierstrass ring, named by harvtxt|Nagata|1962|loc=section 45 after Karl Weierstrass, is a commutative local ring that is Henselian, pseudo geometric, and such that any quotient ring by a prime ideal is a finite extension of a… … Wikipedia
Hensel's lemma — In mathematics, Hensel s lemma, named after Kurt Hensel, is a generic name for analogues for complete commutative rings (including p adic fields in particular) of the Newton method for solving equations. Since p adic analysis is in some ways… … Wikipedia
List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… … Wikipedia
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. It … Wikipedia
Complete field — In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the p adic numbers). See also… … Wikipedia
Hensel — is a surname and may refer to:* Abigail and Brittany Hensel ( Hensel twins ) (1990 ) * Albert Hensel * Bruce Hensel * Karen Hensel * Kurt Hensel (1861 1941), German mathematician ** Hensel s lemma and Henselian rings, named after Kurt * Luise… … Wikipedia
Integral element — In commutative algebra, an element b of a commutative ring B is said to be integral over its subring A if there are such that That is to say, b is a root of a monic polynomial over A.[1] If B consists of elements that are integral over A, then B… … Wikipedia
Nicolae Popescu — Nicolae Popescu, Ph.D., D.Phil. Born 22 September 1937(1937 09 22) Strehaia, Romania Died 29 July 2010(201 … Wikipedia
p-adically closed field — In mathematics, a p adically closed field is a field that enjoys a closure property that is a close analogue for p adic fields to what real closure is to the real field. They were introduced by James Ax and Simon B. Kochen in 1965.[1] Contents 1… … Wikipedia

Academic Dictionaries and Encyclopedias

Henselian ring

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Henselian ring

Look at other dictionaries:

Share the article and excerpts

Direct link