Infinite Galois theory for commutative rings

Infinite Galois Theory

Infinite Products of Zero-dimensional Commutative Rings

On Abelian Hopf Galois structures and finite commutative nilpotent rings

Let G be an elementary abelian p-group of rank n, with p an odd prime. In order to count the Hopf Galois structures of type G on a Galois extension of fields with Galois group G, we need to determine the orbits under conjugation by Aut(G) of regular subgroups of the holomorph of G that are isomorphic to G. The orbits correspond to isomorphism types of commutative nilpotent Fp-algebras N of dime...

On the Galois Theory of Division Rings

1. Throughout this paper, K will represent a division ring and L a galois division subring. We are interested in establishing a galois theory for the extension K/L when K/L is locally finite. In order to do this one must identify the galois subrings of K containing L. An example given by Jacobson [4] shows that not every such division subring is galois. However, we obtain that each subring subj...

ON COMMUTATIVE GELFAND RINGS

A ring is called a Gelfand ring (pm ring ) if each prime ideal is contained in a unique maximal ideal. For a Gelfand ring R with Jacobson radical zero, we show that the following are equivalent: (1) R is Artinian; (2) R is Noetherian; (3) R has a finite Goldie dimension; (4) Every maximal ideal is generated by an idempotent; (5) Max (R) is finite. We also give the following resu1ts:an ideal...