On Galois extensions over commutative rings

Yasuji Takeuchi
In [2], M. Auslander and O. Goldman introduced the notion of Galois extension of commutative rings. Recently, in [5], S. U. Chase, D. K. Harrison and A. Rosenberg gave a generalization of the fundamental theorem of Galois theory. In the first section of this note, we shall extend to a case of noncommutative rings the definition of Galois extension which is defined by Chase, Harrison and Rosenberg in the case of commutative rings. Then we shall establish a half of the fundamental theorem of
more » ... s theory by the method that is completely similar to the method used by Chase, Harrison and Rosenberg. In the second section, we shall study on a Galois extension over commutative rings and we shall show that if a ring Γ is an inner Galois extension of its center C, then Γ is generated by units of Γ over C.
doi:10.18910/4803 fatcat:jzcwyxpbijftnp5teuf4psffoq