A note on Galois theory of commutative rings

Abstract

In [4], S. U. Chase, D. K. Harrison, and Alex Rosenberg succeeded in constructing a finite Galois theory of commutative rings. This paper is about an infinite Galois extension of commutative rings, for which we shall present a corresponding generalization of the Fundamental Theorem of Galois theory. The main results of this paper have been announced in Notices Amer. Math. Soc. 11 (1964), 569. Recently, from Dr. G. J. Janusz we heard that he has obtained many important results for separable algebras over commutative rings, and so in our study we shall refer to his paper [5] to appear, too. Throughout the present paper, S will be a ring with the identity element 1, and R a subring of S containing the identity element 1 of S. As to notations and terminologies used in this paper we follow [2] and [4]. Recently, 0. Villamayor has proved the following theorem which is useful in our study, and he kindly permitted me to cite the proof here.