Partial group actions and partial Galois extensions

2016 ◽  
Vol 185 (2) ◽  
pp. 287-306 ◽  
Author(s):  
Jung-Miao Kuo ◽  
George Szeto
Keyword(s):  
Group Actions ◽  
Galois Extensions ◽  
Partial Group

GALOIS CORRESPONDENCES FOR PARTIAL GALOIS AZUMAYA EXTENSIONS

2011 ◽  
Vol 10 (05) ◽  
pp. 835-847 ◽  
Author(s):  
ANTONIO PAQUES ◽  
VIRGÍNIA RODRIGUES ◽  
ALVERI SANT'ANA
Keyword(s):  
Finite Group ◽  
Galois Theory ◽  
Group Actions ◽  
Galois Extensions ◽  
Partial Action ◽  
Partial Group ◽  
Azumaya Algebra ◽  
Unital Ring ◽  
One To One ◽  
A Finite Group

Let α be a partial action, having globalization, of a finite group G on a unital ring R. Let Rα denote the subring of the α-invariant elements of R and CR(Rα) the centralizer of Rα in R. In this paper we will show that there are one-to-one correspondences among sets of suitable separable subalgebras of R, Rα and CR(Rα). In particular, we extend to the setting of partial group actions similar results due to DeMeyer [Some notes on the general Galois theory of rings, Osaka J. Math.2 (1965) 117–127], and Alfaro and Szeto [On Galois extensions of an Azumaya algebra, Commun. Algebra25 (1997) 1873–1882].

THE EXISTENCE OF CONVEXITY IN PARTIAL GROUP ACTIONS

2021 ◽  
Vol 14 (1) ◽  
pp. 45-55
Author(s):  
S. A. Adebisi ◽  
M. Enioluwafe
Keyword(s):  
Group Actions ◽  
Partial Group

Morita equivalence of partial group actions and globalization

2015 ◽  
Vol 368 (7) ◽  
pp. 4957-4992 ◽  
Author(s):  
F. Abadie ◽  
M. Dokuchaev ◽  
R. Exel ◽  
J. J. Simón
Keyword(s):  
Group Actions ◽  
Morita Equivalence ◽  
Partial Group

Inverse semigroup homomorphisms via partial group actions

2001 ◽  
Vol 64 (1) ◽  
pp. 157-168 ◽  
Author(s):  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Group Actions ◽  
Inverse Semigroups ◽  
Inverse Monoid ◽  
Partial Group ◽  
Inverse Monoids ◽  
Maximal Group ◽  
Special Cases ◽  
Quick Proof ◽  
Group Component

This papar constructs all homomorphisms of inverse semigroups which factor through an E-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domain I of β E-unitary; moreover, the P-representation of I is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is E-unitary. Stronger results are obtained for the case of F-inverse monoids.Special cases of our results include the P-theorem and the factorisation theorem for homomorphisms from E-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence of E-unitary covers over a group, as well as a generalisation to F-inverse covers, allowing a quick proof that every inverse monoid has an F-inverse cover.

Morita Contexts and Partial Group Galois Extensions for Hopf Group Coalgebras

2015 ◽  
Vol 43 (3) ◽  
pp. 1025-1049 ◽  
Author(s):  
Shuang-jian Guo ◽  
Shuan-hong Wang
Keyword(s):  
Galois Extensions ◽  
Partial Group

Partial group actions on algebras

2013 ◽  
Vol 7 ◽  
pp. 517-526
Author(s):  
Ram Parkash Sharma ◽  
Anu
Keyword(s):  
Group Actions ◽  
Partial Group

SEMIALGEBRAS AND THEIR ALGEBRAS OF DIFFERENCES WITH PARTIAL GROUP ACTIONS ON THEM

2013 ◽  
Vol 06 (03) ◽  
pp. 1350038
Author(s):  
Ram Parkash Sharma ◽  
Anu
Keyword(s):  
Finite Group ◽  
Prime Ideal ◽  
Group Actions ◽  
Rocky Mountain ◽  
Prime Ideals ◽  
Central Idempotent ◽  
Group Rings ◽  
Partial Action ◽  
Partial Group ◽  
A Finite Group

Let A be a semialgebra defined in [R. P. Sharma, Anu and N. Singh, Partial group actions on semialgebras, Asian European J. Math.5(4) (2012), Article ID:1250060, 20pp.] over an additively cancellative and commutative semiring K. In additively cancellative semirings, the subtractive ideals play an important role. If P is a subtractive and G-prime ideal of an additively cancellative and yoked semiring A, where G is a finite group acting on A, then A has finitely many n(≤ |G|) minimal primes over P (see [R. P. Sharma and T. R. Sharma, G-prime ideals in semirings and their skew group rings, Comm. Algebra34 (2006) 4459–4465], Lemma 3.6, a result analogous to Lemma 3.2 of Passman [D. S. Passman, It's essentially Maschke's theorem, Rocky Mountain J. Math.13 (1983) 37–54]). Consider a subtractive partial action α of a finite group G on A such that each Dg is generated by a central idempotent 1g of A and the intersection D = ⋂g∈G,Dg≠0Dg of nonzero Dg's is nonzero. It is not necessary that number of minimal primes in Spec A over a subtractive and α-prime ideal P of a yoked semiring A is less than or equal to the order of the group, if 1d ∈ P (Example 3.2). However, we show that the result is true if 1d ∉ P (Corollary 3.1). We also study the prime ideals of the partial fixed subsemiring Aα of A.

PARTIAL GROUP ACTIONS ON SEMIALGEBRAS

2012 ◽  
Vol 05 (04) ◽  
pp. 1250060 ◽  
Author(s):  
Ram Parkash Sharma ◽  
Anu ◽  
Nirmal Singh
Keyword(s):  
Tensor Product ◽  
Group Actions ◽  
Partial Action ◽  
Partial Group

For defining a K-semialgebra A, we use Katsov's tensor product which makes the category K-Smod monoidal. Further, if A is a K-semialgebra then AΔ is a KΔ-algebra and A embeds in AΔ. The subtractive and strong partial actions of a group are defined on A. A subtractive partial action α of a group G on A can be extended to a partial action of G on AΔ which helps in globalization of α. A strong partial action on A has a unique subtractive globalization. We also discuss the associativity of the skew group semiring A ×α G.

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