Galois theory of commutative rings

1992 ◽  
pp. 1-31
Author(s):  
Cornelius Greither
Keyword(s):  
Galois Theory ◽  
Commutative Rings

scholarly journals Galois Theory with Infinitely Many Idempotents

1969 ◽  
Vol 35 ◽  
pp. 83-98 ◽  
Author(s):  
O.E. Villamayor ◽  
D. Zelinsky
Keyword(s):  
Finite Group ◽  
Fundamental Theorem ◽  
Galois Theory ◽  
Linear Independence ◽  
Commutative Rings ◽  
Classical Field ◽  
Fixed Field ◽  
Noncommutative Rings ◽  
Real Change ◽  
A Finite Group

In 1942 Artin proved the linear independence, over a field S, of distinct automorphism of S; in other words if G is a finite group of automorphisms of S and R is the fixed field, then Horn^S, S) is a free S-module with G as basis. Since then, this last condition (“S is G-Galois”) or its equivalents have been used as a postulate in all the Galois theories of rings that are not fields, for example by Dieudonné, Jacobson, Azumaya and Nakayama for noncommutative rings and then in [AG, Appendix] and [CUR] for commutative rings. When S has no idempotents but 0 and 1, [CHR] proves that the ordinary fundamental theorem of Galois theory holds with no real change from the classical, field case.

On the Galois Theory of Commutative Rings II: Automorphisms induced in Residue Rings

1987 ◽  
Vol 39 (5) ◽  
pp. 1025-1037
Author(s):  
Carl Faith
Keyword(s):  
Commutative Ring ◽  
Galois Theory ◽  
Commutative Rings ◽  
Group Of Automorphisms ◽  
Residue Ring ◽  
Image Position

Let G be a group of automorphisms of a commutative ring K, and let KG denote the Galois subring consisting of all elements left fixed by every g in G. An ideal M is G-stable, or G-invariant, provided that g(x) lies in M for every x in M, that is, g(M) ⊆ M, for every g in G. Then, every g in G induces an automorphism in the residue ring , and if is the group consisting of all , trivially1When the inclusion (1) is strict, then G is said to be cleft at M, or by M, and otherwise G is uncleft at (by) M. When G is cleft at all ideals except 0, then G is cleft, and uncleft otherwise.

Admissible Galois Structures on the categories dual to some varieties of universal algebras

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dali Zangurashvili
Keyword(s):  
Galois Theory ◽  
Amalgamation Property ◽  
Commutative Rings ◽  
Free Products ◽  
Linear Ordinary Differential Equations ◽  
Universal Algebras ◽  
Algebraic Version ◽  
Amalgamated Free Products ◽  
Galois Structure ◽  
Homogeneous Linear

AbstractThe subject of the paper is suggested by G. Janelidze and motivated by his earlier result giving a positive answer to the question posed by S. MacLane whether the Galois theory of homogeneous linear ordinary differential equations over a differential field (which is Kolchin–Ritt theory and an algebraic version of Picard–Vessiot theory) can be obtained as a particular case of G. Janelidze’s Galois theory in categories. One ground category in the Galois structure involved in this theory is dual to the category of commutative rings with unit, and another one is dual to the category of commutative differential rings with unit. In the present paper, we apply the general categorical construction, the particular case of which gives this Galois structure, by replacing “commutative rings with unit” by algebras from any variety \mathscr{V} of universal algebras satisfying the amalgamation property and a certain condition (of the syntactical nature) for elements of amalgamated free products which was introduced earlier, and replacing “commutative differential rings with unit” by \mathscr{V}-algebras equipped with additional unary operations which satisfy some special identities to construct a new Galois structure. It is proved that this Galois structure is admissible. Moreover, normal extensions with respect to it are characterized in the case where \mathscr{V} is any of the following varieties: abelian groups, loops and quasigroups.

scholarly journals A Galois theory of commutative rings

2005 ◽  
Vol 289 (2) ◽  
pp. 380-411 ◽  
Author(s):  
David J. Winter
Keyword(s):  
Galois Theory ◽  
Commutative Rings

scholarly journals Infinite Galois theory for commutative rings

1976 ◽  
Vol 64 (1) ◽  
pp. 107-117 ◽  
Author(s):  
John Cruthirds
Keyword(s):  
Galois Theory ◽  
Commutative Rings
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