Jacobson-Bourbaki Correspondence Theorem for Noncommutative Rings

In algebra, the Jacobson-Bourbaki theorem is especially useful for generalizations of the Galois theory of finite, normal and separable field extensions. It was obtained by Jacobson for fields and extended to division rings by Jacobson and Cartan who credited the result to unpublished work by Bourbaki. In 2005, the Jacobson-Bourbaki correspondence theorem for commutative rings was formulated by Winter. And this theorem for augmented rings was formulated by Kadison in 2012. In this paper, we prove the Jacobson-Bourbaki theorem for noncommutative rings which is finitely generated over their centers. We establish a bijective correspondence between the set of subdivisions which are right finite codimension in A and the set of Galois rings of the additive endomorphisms End A of A which is finitely generated over its center.