Galois ring extension

Is there an analogous theory to Galois extension of fields for commutative rings? In particular, what does it mean for a ring extension to be Galois? Thanks.

Solution 1:

In several algebra books there are chapters devoted to integral ring extensions. (I refer to Lang's Algebra, Chapter VII "Extensions of Rings".) Here there are discribed "integrally closed" rings $A$ and there field of fractions $K$, together with a Galois extension $L$ over $K$ and the integral closure $B$ of $A$ in $L$. Instead of the roots of irreducible polynomials in $K$, that are split in $L$, the automorphisms of the Galois group now permute prime ideals of $B$ lying above a fixed prime ideal in $A$.

These kind of ring extensions have several more interesting properties. For details and the definitions of mentioned notions are given in the cited text.

Solution 2:

See the following paper for an accessible introduction, and see also the answers to the MO question Is there a Galois correspondence for ring extensions?

M. Ferrero; A. Paques. Galois Theory of Commutative Rings Revisited.
Contributions to Algebra and Geometry, 38 (1997), No. 2, 399-410.

Solution 3:

You should check Galois Theories of Francis Borceux.