Galois theory of rings

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

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