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A commutative local ring to which the [[Hensel lemma|Hensel lemma]] applies, or, according to another definition, to which the implicit function theorem applies. For a local ring  $  A $
with maximal ideal  $  \mathfrak m $
the former definition means that for any unitary polynomial  $  P( X) \in A[ X] $
and a simple solution  $  a _ {0} \in A $
of the equation  $  P( X) = 0 $(
$   \mathop{\rm mod}  \mathfrak m $) 
(i.e.  $  P ( a _ {0} ) \in \mathfrak m $
and  $  P  ^  \prime  ( a _ {0} ) \notin \mathfrak m $) 
there exists an  $  a \in A $
such that  $  P( a) = 0 $
and  $  a \equiv a _ {0} $(
$   \mathop{\rm mod}  \mathfrak m $).

Examples of Hensel rings include complete local rings, rings of convergent power series (and, in a general sense, analytic rings, cf. [[Analytic ring|Analytic ring]]), and the ring of algebraic power series (i.e. series from  $  k [[ X _ {1} \dots X _ {n} ]] $
which are algebraic over  $  k[ X _ {1} \dots X _ {n} ] $). 
A local ring that is integral over a Hensel ring is a Hensel ring; in particular, a quotient ring of a Hensel ring is a Hensel ring. For any local ring  $  A $
there exists a general construct — a local Hensel  $  A $-
algebra  $  {}  ^ {h} A $
such that for any local Hensel  $  A $-
algebra  $  B $
there exists a unique homomorphism of  $  A $-
algebras  $  {}  ^ {h} A \rightarrow B $. 
The algebra  $  {}  ^ {h} A $
of a local ring  $  A $
is a strictly-flat  $  A $-
module,  $  \mathfrak m {}  ^ {h} A $
will be a maximal ideal of  $  {}  ^ {h} A $, 
the residue fields of  $  A $
and  $  {}  ^ {h} A $
are canonically isomorphic, and the completions of  $  A $
and  $  {}  ^ {h} A $(
in the topologies of the local rings) coincide. Thus, the ring of algebraic power series in  $  X _ {1} \dots X _ {n} $
is a Hensel  $  A $-
algebra for  $  {k [ X _ {1} \dots X _ {n} ] } _ {( X _ {1}  \dots X _ {n} ) } $. 
If  $  A $
is a Noetherian (or, respectively, reduced, normal, regular, excellent) ring, so is  $  {}  ^ {h} A $. 
Conversely, if  $  A $
is an integral ring,  $  {}  ^ {h} A $
need not be integral; more exactly, there exists a bijective correspondence between the maximal ideals of the integral closure of  $  A $
and the minimal prime ideals of  $  {}  ^ {h} A $.

A Hensel ring with a separably-closed residue field is called strictly local (or strictly Henselian), owing to the locality of its spectrum in the étale topology of schemes; in a manner similar to the construction of the Hensel  $  A $-
algebra  $  {}  ^ {h} A $
there is a strict Hensel  $  A $-
algebra functor  $  {}  ^ {sh} A $. 
The concept of a Hensel ring may be introduced for a semi-local ring and even, in a more-general sense, for the pair ring–ideal.

A Hensel ring may be described as a ring over which any finite algebra is a direct sum of local rings. Hensel rings were introduced in [[#References|[1]]]; the general theory of Hensel rings and the construction of Hensel  $  A $-
algebras are developed in [[#References|[2]]].

In the theory of étale morphisms and étale topology a Hensel  $  A $-
algebra is understood to be the inductive limit of étale extensions of the ring. In a commutative algebra a Hensel  $  A $-
algebra often replaces the operation of completion, which plays an important role in local studies of objects.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Azumaya,   "On maximally central algebras"  ''Nagoya Math. J.'' , '''2'''  (1951)  pp. 119–150</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Nagata,   "Local rings" , Interscience  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Grothendieck,   "Eléments de géometrie algébrique. IV"  ''Publ. Math. IHES'' :  32  (1967)</TD></TR></table>

====Comments====
The ring (algebra)  $  {}  ^ {sh} A $
is called the Henselization or Hensel closure of the local ring  $  A $.

The ideal–ring pair formulation of the Hensel property is as follows. Let  $  ( A , I) $
be a pair consisting of a ring  $  A $
and an ideal  $  I $. 
Then if  $  f \in A[ I] $
is such that  $  f( 0) \in I $
and  $  f ^ { \prime } ( 0) $
is a unit in  $  A / I $, 
then there exists an  $  a \in I $
such that  $  f( a) = 0 $.

For a discussion of the solution of systems of polynomial equations and implicit-function type statements in the context of Hensel rings cf., for instance, [[#References|[a2]]], Chapt. 2.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Raynaud,   "Anneaux locaux Henséliens" , ''Lect. notes in math.'' , '''169''' , Springer  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Kurke,   G. Pfister,   M. Roczen,   "Henselsche Ringe" , Deutsch. Verlag Wissenschaft.  (1975)</TD></TR></table>