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A generalization of the results of the theory of Galois fields (cf. [[Galois theory|Galois theory]] and [[Galois group|Galois group]]) to the case of associative rings with a unit element. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g0431801.png" /> be an associative ring with a unit element, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g0431802.png" /> be some subgroup of the group of all automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g0431803.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g0431804.png" /> be a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g0431805.png" />, let
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g0431806.png" /></td> </tr></table>
+
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 +
{{TEX|done}}
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g0431807.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g0431808.png" /> will then be a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g0431809.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318010.png" /> be a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318011.png" />. One says that an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318013.png" /> leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318014.png" /> invariant elementwise if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318015.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318016.png" />. The set of all such automorphisms is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318017.png" />. Let
+
A generalization of the results of the theory of Galois fields (cf. [[Galois theory|Galois theory]] and [[Galois group|Galois group]]) to the case of associative rings with a unit element. Let $  A $
 +
be an associative ring with a unit element, let  $  H $
 +
be some subgroup of the group of all automorphisms of  $  A $,
 +
let  $  N $
 +
be a subgroup of  $  H $,
 +
let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318018.png" /></td> </tr></table>
+
$$
 +
J ( N)  = \
 +
\{ {a \in A } : {h ( a) = a, \forall h \in N } \}
 +
,
 +
$$
 +
 
 +
and let  $  B = J ( H) $.
 +
The set  $  J ( N) $
 +
will then be a subring of  $  A $.
 +
Let  $  B _ {1} $
 +
be a subring of  $  A $.
 +
One says that an automorphism  $  h $
 +
of  $  A $
 +
leaves  $  B _ {1} $
 +
invariant elementwise if  $  h( b) = b $
 +
for all  $  b \in B _ {1} $.  
 +
The set of all such automorphisms is denoted by  $  G( B _ {1} ) $.  
 +
Let
 +
 
 +
$$
 +
H ( B _ {1} )  = \
 +
G ( B _ {1} ) \cap H \ \
 +
\textrm{ and } \ \
 +
B _ {1}  \supseteq  B.
 +
$$
  
 
The principal subject of the Galois theory of rings are the correspondences:
 
The principal subject of the Galois theory of rings are the correspondences:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318019.png" />;
+
1) $  N  \rightarrow  J ( N) $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318020.png" />;
+
2) $  B _ {1}  \rightarrow  G( B _ {1} ) $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318021.png" />. Unlike the Galois theory of fields, (even when the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318022.png" /> is finite) the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318023.png" /> is not always valid, while the correspondences 1), 2) and 1), 3) need not be mutually inverse. It is of interest, accordingly, to single out families of subrings and families of subgroups for which the analogue of the theorem on [[Galois correspondence|Galois correspondence]] is valid. This problem has found a positive solution in two cases. The first one involves the requirements of "proximity" between the properties of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318024.png" /> and the properties of a field (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318025.png" /> is a skew-field or a complete ring of linear transformations of a vector space over a skew-field); the second is the requirement of "proximity" between the structure of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318026.png" /> over a subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318027.png" /> to the structure of the corresponding pair if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318028.png" /> is a field (e.g. the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318029.png" />-module is projective).
+
3) $  B _ {1}  \rightarrow  H ( B _ {1} ) $.  
 +
Unlike the Galois theory of fields, (even when the group $  H $
 +
is finite) the equality $  G( B _ {1} ) = H ( B _ {1} ) $
 +
is not always valid, while the correspondences 1), 2) and 1), 3) need not be mutually inverse. It is of interest, accordingly, to single out families of subrings and families of subgroups for which the analogue of the theorem on [[Galois correspondence|Galois correspondence]] is valid. This problem has found a positive solution in two cases. The first one involves the requirements of "proximity" between the properties of the ring $  A $
 +
and the properties of a field (e.g. $  A $
 +
is a skew-field or a complete ring of linear transformations of a vector space over a skew-field); the second is the requirement of "proximity" between the structure of the ring $  A $
 +
over a subring $  B $
 +
to the structure of the corresponding pair if $  A $
 +
is a field (e.g. the $  B $-module is projective).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318030.png" /> be an invertible element of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318031.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318032.png" /> be the automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318033.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318035.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318036.png" /> be the subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318037.png" /> generated by the invertible elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318038.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318039.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318040.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318042.png" />-group if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318043.png" /> for all invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318044.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318045.png" /> is a skew-field, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318046.png" /> is a sub-skew-field of it, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318047.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318048.png" /> is a finite-dimensional left vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318049.png" />, then the Galois correspondences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318051.png" /> are mutually inverse, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318052.png" /> belongs to the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318053.png" />-subgroups of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318055.png" /> to the set of all skew-fields of the sub-skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318056.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318057.png" />.
+
Let $  c $
 +
be an invertible element of the ring $  A $,  
 +
let $  T _ {c} : A \rightarrow A $
 +
be the automorphism of $  A $
 +
defined by $  T _ {c} ( x) = cxc  ^ {-} 1 $,  
 +
$  x \in A $,  
 +
and let $  R( H) $
 +
be the subalgebra of $  A $
 +
generated by the invertible elements $  c \in A $
 +
for which $  T _ {c} \in H $.  
 +
The group $  H $
 +
is called an $  N $-group if $  T _ {x} \in H $
 +
for all invertible $  x \in R( H) $.  
 +
If $  A $
 +
is a skew-field, if $  B $
 +
is a sub-skew-field of it, if $  B = J( G( B)) $,  
 +
and if $  A $
 +
is a finite-dimensional left vector space over $  B $,  
 +
then the Galois correspondences $  H \rightarrow J( H) $
 +
and $  D \rightarrow G( D) $
 +
are mutually inverse, where $  H $
 +
belongs to the set of all $  N $-subgroups of the group $  G( B) $
 +
and $  D $
 +
to the set of all skew-fields of the sub-skew-field $  A $
 +
containing $  B $.
  
A similar result is also valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318058.png" /> is a complete ring of linear transformations (but the corresponding system of conditions singling out the families of subgroups and families of subrings is formulated in a somewhat more complicated manner).
+
A similar result is also valid if $  A $
 +
is a complete ring of linear transformations (but the corresponding system of conditions singling out the families of subgroups and families of subrings is formulated in a somewhat more complicated manner).
  
Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318059.png" /> be a commutative ring without non-trivial idempotents and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318060.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318061.png" /> is called a finite normal extension of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318062.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318064.png" /> is a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318065.png" />-module. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318066.png" /> may be considered to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318067.png" />-module by assuming
+
Further, let $  A $
 +
be a commutative ring without non-trivial idempotents and let $  A \supset B $.  
 +
The ring $  A $
 +
is called a finite normal extension of a ring $  B $
 +
if $  B = J( G( B)) $
 +
and $  A $
 +
is a finitely-generated $  B $-module. The ring $  A $
 +
may be considered to be an $  A \otimes _ {B} A $-module by assuming
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318068.png" /></td> </tr></table>
+
$$
 +
\left (
 +
\sum _ {i = 1 } ^ { n }
 +
a _ {i} \otimes b _ {i} \right ) a  = \
 +
\sum _ {i = 1 } ^ { n }
 +
a _ {i} b _ {i} a,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318069.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318070.png" /> is called a separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318072.png" />-algebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318073.png" /> is a projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318074.png" />-module. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318075.png" /> is a finite normal separable extension of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318076.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318077.png" /> is a finitely-generated projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318078.png" />-module, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318079.png" /> is finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318080.png" /> and the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318082.png" /> define mutually-inverse relations between the set of all subgroups of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318083.png" /> and the set of all separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318084.png" />-subalgebras of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318085.png" />.
+
where $  a _ {i} , b _ {i} , a \in A $.  
 +
The ring $  A $
 +
is called a separable $  B $-algebra if $  A $
 +
is a projective $  A \otimes _ {B} A $-module. If $  A $
 +
is a finite normal separable extension of the ring $  B $,  
 +
then $  A $
 +
is a finitely-generated projective $  B $-module, the group $  G( B) $
 +
is finite $  ([ G( B) : 1] = \mathop{\rm rank} _ {B}  A ) $
 +
and the mappings $  H \rightarrow J( H) $,  
 +
$  B _ {1} \rightarrow G( B _ {1} ) $
 +
define mutually-inverse relations between the set of all subgroups of the group $  G( B) $
 +
and the set of all separable $  B $-subalgebras of the algebra $  A $.
  
Any ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318086.png" /> has a separable closure, which is an analogue of the separable closure of a field. The group of all automorphisms of this closure which leave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318087.png" /> invariant elementwise is, in the general case, a profinite group. The correspondences 1) and 2) are mutually inverse on the set of all closed subgroups of the resulting group and on the set of all separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318088.png" />-subalgebras of the separable closure of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318089.png" />.
+
Any ring $  B $
 +
has a separable closure, which is an analogue of the separable closure of a field. The group of all automorphisms of this closure which leave $  B $
 +
invariant elementwise is, in the general case, a profinite group. The correspondences 1) and 2) are mutually inverse on the set of all closed subgroups of the resulting group and on the set of all separable $  B $-subalgebras of the separable closure of the ring $  B $.
  
Similar results are also valid if the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318090.png" /> contains non-trivial idempotents. However, this involves substantial changes in a number of basic concepts. For instance, the role of the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318091.png" /> is taken over by the [[Fundamental groupoid|fundamental groupoid]].
+
Similar results are also valid if the ring $  B $
 +
contains non-trivial idempotents. However, this involves substantial changes in a number of basic concepts. For instance, the role of the Galois group $  G( B) $
 +
is taken over by the [[Fundamental groupoid|fundamental groupoid]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) {{MR|0081264}} {{ZBL|0073.02002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.U. Chase, M.E. Swedler, "Hopf algebras and Galois theory" , Springer (1969) {{MR|0260724}} {{ZBL|0197.01403}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> F. de Meyer, E. Ingraham, "Separable algebras over commutative rings" , ''Lect. notes in math.'' , '''181''' , Springer (1971) {{MR|280479}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.R. Magid, "The separable Galois theory of commutative rings" , M. Dekker (1974) {{MR|0352075}} {{ZBL|0284.13004}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) {{MR|0081264}} {{ZBL|0073.02002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.U. Chase, M.E. Swedler, "Hopf algebras and Galois theory" , Springer (1969) {{MR|0260724}} {{ZBL|0197.01403}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> F. de Meyer, E. Ingraham, "Separable algebras over commutative rings" , ''Lect. notes in math.'' , '''181''' , Springer (1971) {{MR|280479}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.R. Magid, "The separable Galois theory of commutative rings" , M. Dekker (1974) {{MR|0352075}} {{ZBL|0284.13004}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.U. Chase, D.K. Harrison, A. Rosenberg, "Galois theory and Galois cohomology of commutative rings" , ''Mem. Amer. Math. Soc.'' , '''52''' , Amer. Math. Soc. (1965) {{MR|0195922}} {{ZBL|0143.05902}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.U. Chase, D.K. Harrison, A. Rosenberg, "Galois theory and Galois cohomology of commutative rings" , ''Mem. Amer. Math. Soc.'' , '''52''' , Amer. Math. Soc. (1965) {{MR|0195922}} {{ZBL|0143.05902}} </TD></TR></table>

Latest revision as of 04:16, 19 March 2022


A generalization of the results of the theory of Galois fields (cf. Galois theory and Galois group) to the case of associative rings with a unit element. Let $ A $ be an associative ring with a unit element, let $ H $ be some subgroup of the group of all automorphisms of $ A $, let $ N $ be a subgroup of $ H $, let

$$ J ( N) = \ \{ {a \in A } : {h ( a) = a, \forall h \in N } \} , $$

and let $ B = J ( H) $. The set $ J ( N) $ will then be a subring of $ A $. Let $ B _ {1} $ be a subring of $ A $. One says that an automorphism $ h $ of $ A $ leaves $ B _ {1} $ invariant elementwise if $ h( b) = b $ for all $ b \in B _ {1} $. The set of all such automorphisms is denoted by $ G( B _ {1} ) $. Let

$$ H ( B _ {1} ) = \ G ( B _ {1} ) \cap H \ \ \textrm{ and } \ \ B _ {1} \supseteq B. $$

The principal subject of the Galois theory of rings are the correspondences:

1) $ N \rightarrow J ( N) $;

2) $ B _ {1} \rightarrow G( B _ {1} ) $;

3) $ B _ {1} \rightarrow H ( B _ {1} ) $. Unlike the Galois theory of fields, (even when the group $ H $ is finite) the equality $ G( B _ {1} ) = H ( B _ {1} ) $ is not always valid, while the correspondences 1), 2) and 1), 3) need not be mutually inverse. It is of interest, accordingly, to single out families of subrings and families of subgroups for which the analogue of the theorem on Galois correspondence is valid. This problem has found a positive solution in two cases. The first one involves the requirements of "proximity" between the properties of the ring $ A $ and the properties of a field (e.g. $ A $ is a skew-field or a complete ring of linear transformations of a vector space over a skew-field); the second is the requirement of "proximity" between the structure of the ring $ A $ over a subring $ B $ to the structure of the corresponding pair if $ A $ is a field (e.g. the $ B $-module is projective).

Let $ c $ be an invertible element of the ring $ A $, let $ T _ {c} : A \rightarrow A $ be the automorphism of $ A $ defined by $ T _ {c} ( x) = cxc ^ {-} 1 $, $ x \in A $, and let $ R( H) $ be the subalgebra of $ A $ generated by the invertible elements $ c \in A $ for which $ T _ {c} \in H $. The group $ H $ is called an $ N $-group if $ T _ {x} \in H $ for all invertible $ x \in R( H) $. If $ A $ is a skew-field, if $ B $ is a sub-skew-field of it, if $ B = J( G( B)) $, and if $ A $ is a finite-dimensional left vector space over $ B $, then the Galois correspondences $ H \rightarrow J( H) $ and $ D \rightarrow G( D) $ are mutually inverse, where $ H $ belongs to the set of all $ N $-subgroups of the group $ G( B) $ and $ D $ to the set of all skew-fields of the sub-skew-field $ A $ containing $ B $.

A similar result is also valid if $ A $ is a complete ring of linear transformations (but the corresponding system of conditions singling out the families of subgroups and families of subrings is formulated in a somewhat more complicated manner).

Further, let $ A $ be a commutative ring without non-trivial idempotents and let $ A \supset B $. The ring $ A $ is called a finite normal extension of a ring $ B $ if $ B = J( G( B)) $ and $ A $ is a finitely-generated $ B $-module. The ring $ A $ may be considered to be an $ A \otimes _ {B} A $-module by assuming

$$ \left ( \sum _ {i = 1 } ^ { n } a _ {i} \otimes b _ {i} \right ) a = \ \sum _ {i = 1 } ^ { n } a _ {i} b _ {i} a, $$

where $ a _ {i} , b _ {i} , a \in A $. The ring $ A $ is called a separable $ B $-algebra if $ A $ is a projective $ A \otimes _ {B} A $-module. If $ A $ is a finite normal separable extension of the ring $ B $, then $ A $ is a finitely-generated projective $ B $-module, the group $ G( B) $ is finite $ ([ G( B) : 1] = \mathop{\rm rank} _ {B} A ) $ and the mappings $ H \rightarrow J( H) $, $ B _ {1} \rightarrow G( B _ {1} ) $ define mutually-inverse relations between the set of all subgroups of the group $ G( B) $ and the set of all separable $ B $-subalgebras of the algebra $ A $.

Any ring $ B $ has a separable closure, which is an analogue of the separable closure of a field. The group of all automorphisms of this closure which leave $ B $ invariant elementwise is, in the general case, a profinite group. The correspondences 1) and 2) are mutually inverse on the set of all closed subgroups of the resulting group and on the set of all separable $ B $-subalgebras of the separable closure of the ring $ B $.

Similar results are also valid if the ring $ B $ contains non-trivial idempotents. However, this involves substantial changes in a number of basic concepts. For instance, the role of the Galois group $ G( B) $ is taken over by the fundamental groupoid.

References

[1] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) MR0081264 Zbl 0073.02002
[2] S.U. Chase, M.E. Swedler, "Hopf algebras and Galois theory" , Springer (1969) MR0260724 Zbl 0197.01403
[3] F. de Meyer, E. Ingraham, "Separable algebras over commutative rings" , Lect. notes in math. , 181 , Springer (1971) MR280479
[4] A.R. Magid, "The separable Galois theory of commutative rings" , M. Dekker (1974) MR0352075 Zbl 0284.13004

Comments

References

[a1] S.U. Chase, D.K. Harrison, A. Rosenberg, "Galois theory and Galois cohomology of commutative rings" , Mem. Amer. Math. Soc. , 52 , Amer. Math. Soc. (1965) MR0195922 Zbl 0143.05902
How to Cite This Entry:
Galois theory of rings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galois_theory_of_rings&oldid=24073
This article was adapted from an original article by K.I. BeidarA.V. Mikhalev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article