View source for Galois theory of rings

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

<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>

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

<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>

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" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043180/g04318020.png" />;

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).

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" />.

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).

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

<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>

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" />.

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" />.

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]].

====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>



====Comments====


====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>