Partial Galois Extension Research Articles

On Hirata separability and inner partial group action

In this paper, we generalize in the context of partial group action a result of Sugano by giving some equivalent conditions for partial Galois extensions to be Hirata-separable extensions. As an application, we show that every partial Galois extension with inner partial group action is Hirata-separable.

On the induced partial action of a quotient group and a structure theorem for a partial Galois extension

Let [Formula: see text] be a ring with a partial action [Formula: see text] of a finite group [Formula: see text]. We determine when a quotient group of [Formula: see text] gives rise to a partial action induced by [Formula: see text] on a subring of [Formula: see text]. As an application, we show that if [Formula: see text] is an [Formula: see text]-partial Galois extension and [Formula: see text] is a normal subgroup of [Formula: see text], then under certain conditions [Formula: see text] under the partial action of [Formula: see text] induced by [Formula: see text], denoted by [Formula: see text], is a partial Galois extension. This result was just shown recently by Bagio et al., applying globalization to derive a partial action of [Formula: see text] on [Formula: see text], totally different from the one presented in this paper arising from a Boolean ring generated by certain idempotents. We further show in either type of induced partial action of a quotient group that under certain conditions [Formula: see text] is a DeMeyer–Kanzaki [Formula: see text]-partial Galois extension whenever [Formula: see text] is an [Formula: see text]-partial Galois Azumaya extension. A structure theorem for a partial Galois extension is also presented, namely, every partial Galois extension can be decomposed as a direct sum of Galois extensions and possibly a trivial partial Galois extension of type II.

The commutative inverse semigroup of partial abelian extensions

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an -partial Galois extension of and a normal subgroup H of G, we prove that induces a unital partial action of G/H on the subalgebra of invariants of S such that is an -partial Galois extension of Second, assuming that G is abelian, we construct a commutative inverse semigroup whose elements are equivalence classes of -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between and T(G, A), where ρ is a congruence on and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of reduces to the case where G is cyclic. The set of idempotents of is also investigated.

Partial generalized crossed products and a seven-term exact sequence

For a partial Galois extension of commutative rings we give a seven term exact sequence which generalize the Chase-Harrison-Rosenberg sequence.

Galois corings and groupoids acting partially on algebras

Bagio and Paques [Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra 40 (2012) 3658–3678] developed a Galois theory for unital partial actions by finite groupoids. The aim of this note is to show that this is actually a special case of the Galois theory for corings, as introduced by Brzeziński [The structure of corings, Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Represent. Theory 5 (2002) 389–410]. To this end, we associate a coring to a unital partial action of a finite groupoid on an algebra [Formula: see text], and show that this coring is Galois if and only if [Formula: see text] is an [Formula: see text]-partial Galois extension of its coinvariants.

On three special types of partial Galois extensions

In this paper, we study the following three special types of partial Galois extensions: DeMeyer–Kanzaki partial Galois extension, partial Galois Azumaya extension and commutator partial Galois extension.

Partial Galois cohomology and related homomorphisms

Abstract For a partial Galois extension of commutative rings, we give a seven terms sequence, which is an analog of the Chase–Harrison–Rosenberg sequence.

Partial group actions and partial Galois extensions

Let \(\alpha \) be a partial action of a group G on a ring S which has an enveloping action. Suppose that \((S,\alpha )\) is a partial Galois extension. We study partial Galois extensions inside \((S,\alpha )\). In particular, we derive some results on partial orbits and partial stabilizers and apply them to associate to each subgroup K of G certain partial Galois extensions inside \((S,\alpha )\) with partial actions of \(\alpha \) restricted to K.

The structure of a partial Galois extension II

Let [Formula: see text] be a partial Galois extension where [Formula: see text] is a partial action of a finite group on a ring [Formula: see text] such that the associated ideals are generated by central idempotents. We determine the set of all Galois extensions in [Formula: see text], and give an orthogonality criterion for nonzero elements in the Boolean semigroup generated by those central idempotents. These results lead to a structure theorem for [Formula: see text].

The group idempotents in a partial Galois extension

Let (R, αG) be a partial Galois extension with a partial action αG of a finite group G, B the Boolean ring generated by {1g | g ∈ G} where 1g is the central idempotent associated with g ∈ G. Let e ≠ 0 ∈ B and G(e) = {g ∈ G | e1g ≠ 0}. We call e a group idempotent if G(e) is a subgroup of G. It is shown that if e is a group idempotent, then (Re, αG(e)) is a partial Galois extension induced by e. Thus the set of these partial Galois extensions in (R, αG) is computed, and a structure theorem for (R, αG) is obtained.

Morita Contexts and Partial Group Galois Extensions for Hopf Group Coalgebras

We first introduce the notion of partial group twisted smash product and construct a Morita context for partial coaction of co-Frobenius Hopf group coalgebra relating generalized partial group smash product and partial coinvariants. Furthermore, we study group-Galois extensions and reobtain some classical results in the partial case. Finally, we prove that any partial group Galois extension induces a unique partial group entwining map compatible with the partial right coaction.

Galois extensions induced by a central idempotent in a partial Galois extension

Let (R,α) be a partial Galois extension of RαG with a partial action of a finite group G, e a non-zero central idempotent in R, 1g the central idempotent associated with g ∈ G, and E = e(Πg∈G1g) 6= 0 with a maximal number of factors 1g for g ∈ G. A sufficient condition for a Galois extension Re with Galois group H(e) and for a Galois extension RE with Galois group N(e) is given respectively, where H(e) = {g ∈ G|e1g = e} and N(e) = {g ∈ G|e(Πg∈G1g) 6= 0} with a maximal number of factors 1g for g ∈ G. This leads to a structure of Re as a direct sum of Galois extensions. Mathematics Subject Classification: 13B05

Morita context, partial Hopf Galois extensions and partial entwining structure

We first introduce the notion of a right generalized partial smash product and explore some properties of such partial smash product. Based on these notions and properties, then we construct a Morita context for partial coactions of a co-Frobenius Hopf algebra. Finally, we prove that any Hopf partial Galois extension induces a unique partial entwining map compatible with the right partial coaction.

The structure of a partial Galois extension

In this paper, we present an easy way to construct partial Galois extensions; in particular, any direct sum of finitely many Galois extensions forms a partial Galois extension. The idea is inspired by the study of how Galois extensions are embedded in a partial Galois extension via minimal elements in an associated Boolean semigroup.

Partial Groupoid Actions: Globalization, Morita Theory, and Galois Theory

In this article, we introduce the notion of a partial action of a groupoid on a ring as well as we give a criteria for the existence of a globalization of it. We construct a Morita context associated to a globalizable partial groupoid action, and we introduce the notion of a partial Galois extension, which is related to the strictness of this context.