Henselian ring

E483407

algebraic structure property ring theory concept

A Henselian ring is a local ring in which Hensel’s lemma holds, allowing certain types of polynomial factorizations and root liftings from the residue field to the ring itself.

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Statements (46)

Predicate	Object
instanceOf	algebraic structure property ⓘ ring theory concept ⓘ
allows	lifting of certain polynomial factorizations from residue field to the ring ⓘ lifting of simple roots from residue field to the ring ⓘ
appearsIn	study of local behavior of morphisms of schemes ⓘ theory of lifting solutions of polynomial equations ⓘ
contrastsWith	non-Henselian local ring ⓘ
field	commutative algebra ⓘ
generalizes	complete Noetherian local ring ⓘ
hasComponent	maximal ideal ⓘ residue field ⓘ
hasConstruction	henselization of a given local ring ⓘ
hasDefinition	a local ring in which Hensel’s lemma holds ⓘ
hasEquivalentCondition	every finite ring extension decomposes according to its residue field decomposition ⓘ idempotents of any finite algebra over the ring lift from the residue algebra ⓘ
hasExample	Henselization of a local ring ⓘ complete local ring with respect to its maximal ideal ⓘ p-adic integers Z_p ⓘ strictly Henselian local ring ⓘ
hasProperty	Henselian property is invariant under isomorphism of local rings ⓘ Henselian property is local on Spec in the sense of local rings at points ⓘ Henselian property is preserved under integral closure in finite extensions of fraction fields (under suitable hypotheses) ⓘ Henselian property is stable under finite products of local rings ⓘ Hensel’s lemma holds for monic polynomials ⓘ completion of a Noetherian local ring is Henselian ⓘ if a local ring is strictly Henselian then it is Henselian ⓘ if the residue field is separably closed and the ring is Henselian, then the ring is strictly Henselian ⓘ local ring ⓘ maximal ideal is contained in the Jacobson radical ⓘ
hasSubClass	complete local ring ⓘ strictly Henselian ring ⓘ
hasUniversalProperty	henselization is initial among Henselian local rings receiving a local homomorphism from the given ring ⓘ
implies	uniqueness of lifting of simple roots under suitable conditions ⓘ
isCharacterizedBy	every finite algebra over the ring decomposes according to its residue algebra decomposition ⓘ factorizations of monic polynomials over the residue field lift to factorizations over the ring under suitable conditions ⓘ idempotents lift uniquely from residue ring to the ring ⓘ
isNotEquivalentTo	complete local ring in general ⓘ
namedAfter	Kurt Hensel NERFINISHED ⓘ
relatedTo	Henselization ⓘ henselization functor ⓘ étale morphism ⓘ
satisfies	Hensel’s lemma NERFINISHED ⓘ
usedIn	algebraic geometry ⓘ local study of schemes ⓘ number theory ⓘ valuation theory ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Kurt Hensel → developedConcept → Henselian ring ⓘ

Kurt Hensel → hasEponym → Henselian ring ⓘ