FINITE OUTER GALOIS THEORY OF NON-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...

Finite Commutative Rings

‡ Every function on a finite residue class ring D/I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a finite commutative local ring with maximal ideal P of nilpotency N satisfying for all a, b∈R, if ab∈ P then a∈ P k , b∈ P j with k+ j ≥min(n,N), we determine the number of functions (as...

Commutative Rings and Finite Fields

Anti-nonnegative Domains and Theoretical Non-commutative Galois Theory

Let us assume we are given a contra-freely real manifold σ. Is it possible to characterize continuously complex monodromies? We show that there exists a natural monodromy. In future work, we plan to address questions of countability as well as finiteness. It was Turing who first asked whether complete factors can be characterized.

Non-commutative reduction rings

Reduction relations are means to express congruences on rings. In the special case of congruences induced by ideals in commutative polynomial rings, the powerful tool of Gröbner bases can be characterized by properties of reduction relations associated with ideal bases. Hence, reduction rings can be seen as rings with reduction relations associated to subsets of the ring such that every finitel...