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In this paper, our objects of interest are Hopf Galois extensions (e.g., Hopf algebras, Galois field extensions, strongly graded algebras, crossed products, principal bundles, etc.) and families of noncommutative rings (e.g., skew polynomial rings, PBW extensions and skew PBW extensions, etc.) We collect and systematize questions, problems, properties and recent advances in both theories by explicitly developing examples and doing calculations that are usually omitted in the literature. In particular, for Hopf Galois extensions we consider approaches from the point of view of quantum torsors (also known as quantum heaps) and Hopf Galois systems, while for some families of noncommutative rings we present advances in the characterization of ring-theoretic and homological properties. Every developed topic is exemplified with abundant references to classic and current works, so this paper serves as a survey for those interested in either of the two theories. Throughout, interactions between both are presented.

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A skew brace G is a set with two group operations, one defining a (not necessarily abelian) "additive group" on G and the other a "circle group" on G, so that G with the two operations satisfies a relation analogous to distributivity. If G is a skew brace, then G yields a Hopf Galois structure of type equal to the additive group of G on any Galois extension of fields with Galois group isomorphic to the circle group of G. A skew brace G is a bi-skew brace if it is also a skew brace with the roles of the circle and additive group reversed. In that event, then G also corresponds to a Hopf Galois structure of type equal to the circle group on a Galois extension of fields with Galois group isomorphic to the additive group. Many non-trivial examples exist. One source is radical rings A with A³ = 0, where one of the groups is abelian and the other need not be. We find that the left braces of degree p³ classified by Bachiller are bi-skew braces if and only they are radical rings. A different source of bi-skew braces is semidirect products of arbitrary finite groups, which yield many examples where both groups are non- abelian, and a skew brace proof of a result of Crespo, Rio and Vela that if G is a semidirect product of two finite groups H and J, then any Galois extension of fields with Galois group G has a Hopf Galois structure of type equal to the direct product of H and J. [ABSTRACT FROM AUTHOR]

In this paper, the classical Galois theory to the H*-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra, and A/ A a right H*-Galois extension. The authors prove that, if A is a separable k-algebra, then for any right coideal subalgebra B of H, the B-invariants A = { a ∈ A | b · a = ε( b) a, ∀ b ∈ B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing A as in the classical case. The results are applied to the case H = ( kG)* for a finite group G to get a Galois 1-1 correspondence. [ABSTRACT FROM AUTHOR]

We study the question of the surjectivity of the Galois correspondence from subHopf algebras to subfields given by the Fundamental Theorem of Galois Theory for abelian Hopf Galois structures on a Galois extension of fields with Galois group +, a finite abelian p-group. Applying the connection between regular subgroups of the holomorph of a finite abelian p-group (G; +) and associative, commutative nilpotent algebra structures A on (G; +), we show that if A gives rise to a H-Hopf Galois structure on L=K, then the K-subHopf algebras of H correspond to the ideals of A. Among the applications, we show that if G and + are both elementary abelian p-groups, then the only Hopf Galois structure on L=K of type G for which the Galois correspondence is surjective is the classical Galois structure. [ABSTRACT FROM AUTHOR]

LetK/kbe a finite separable extension,nits degree andits Galois closure. Forn ≤ 5, Greither and Pareigis show that all Hopf Galois extensions are either Galois or almost classically Galois and they determine the Hopf Galois character ofK/kaccording to the Galois group (or the degree) of. In this paper we study the casen = 6, and intermediate extensionsF/ksuch that, for degreesn = 4, 5, 6. We present an example of a non almost classically Galois Hopf Galois extension of ℚ of the smallest possible degree and new examples of Hopf Galois extensions. In the last section we prove a transitivity property of the Hopf Galois condition. [ABSTRACT FROM AUTHOR]

Let H be a finite dimensional Hopf algebra over a field k, H* the dual Hopf algebra of H, B a left H-module algebra, and VB(BH) the commutator subring of BH in B. It is shown that VB(BH) is an H*-Galois extension and an Azumaya algebra if and only if the smash product B#H is a Hirata separable extension of B and B is a Hirata separable extension of BH. [ABSTRACT FROM AUTHOR]