Generalized Galois Theory for Rings with Minimum Condition

Abstract

In our first paper, Generalized Galois theory for rings with minimum condition, Amer. J. Math., vol. 73 (1951), pp. 1-12-referred to as I-we gave a certain Galois theory for primary rings with minimum condition. The theory contains the Cartan-Jacobson [1], [2] Galois theory for sfields, at least as far as the main Galois correspondence is concerned. But the inner portion of the Galois group, there considered, comes from a certain completely primary subring. Now that we are dealing with a general primary ring, satisfying the minimum condition, it is desirable to consider a wider class of Galois groups whose inner portions come from general primary subrings rather than from completely primary ones. In fact in-another paper [5] we have developed a Galois theory for simple rings, with minimum condition, in which the Galois groups have rather satisfactory generality and have inner portions consisting of inner automorphisms effected by regular elements of simple subalgebras. Combining the methods and results of this paper with those in I, we now give a Galois theory for primary rings which deals with Galois groups of the generality required above, though it fails to cover the theory of I completely. Our main Galois correspondence is given in 2, Theorem 1, our regular Galois group being defined both at the opening of 2 and in succession to Theorem 1. Moreover, after a study of a certain special type of factor-groups of the Galois group (3, Theorem 2), we prove an extension theorem for isomorphisms (3, Theorem 3), to recover a second main aspect of Galois theory which was neglected in our first paper I.