A Galois theory of commutative rings

Abstract

Galois objects—Galois groups, rings, Lie rings, and birings G —act on commutative rings A and satisfy Galois correspondence theorems which support Galois descent. This generalizes the Galois theory of fields to a Galois theory of commutative rings. In particular, the classical correspondence of Galois, the Jacobson–Bourbaki correspondence [N. Jacobson, Lectures in Abstract Algebra, vol. 3, Van Nostrand, 1964; D.J. Winter, The Jacobson descent theorem, Pacific J. Math. 104 (2) (1983) 495–496; D.J. Winter, The Structure of Fields, Springer-Verlag, 1974], the Jacobson differential correspondence [N. Jacobson, op. cit.; D.J. Winter, The Structure of Fields, op. cit.], the Galois birings correspondence of [D.J. Winter, The Structure of Fields, op. cit.], and corresponding theories of Galois descent [N. Jacobson, Forms of algebras, Yeshiva Sci. Confs. 7 (1966) 41–71; D.J. Winter, The Jacobson descent theorem, op. cit.; D.J. Winter, The Structure of Fields, op. cit.] generalize from fields to commutative rings. The Galois Lie rings correspondence Theorem 4.2 solves the simple restricted irreducible derivation rings Problem 8.4 in the finitely generated case.