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].

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.

The structure of a partial Galois extension II

2016 ◽  
Vol 15 (04) ◽  
pp. 1650061 ◽  
Author(s):  
Jung-Miao Kuo ◽  
George Szeto
Keyword(s):  
Finite Group ◽  
Galois Extension ◽  
Structure Theorem ◽  
Galois Extensions ◽  
Partial Action ◽  
A Finite Group ◽  
Partial Galois Extension

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].

scholarly journals Galois Theory with Infinitely Many Idempotents

1969 ◽  
Vol 35 ◽  
pp. 83-98 ◽  
Author(s):  
O.E. Villamayor ◽  
D. Zelinsky
Keyword(s):  
Finite Group ◽  
Fundamental Theorem ◽  
Galois Theory ◽  
Linear Independence ◽  
Commutative Rings ◽  
Classical Field ◽  
Fixed Field ◽  
Noncommutative Rings ◽  
Real Change ◽  
A Finite Group

In 1942 Artin proved the linear independence, over a field S, of distinct automorphism of S; in other words if G is a finite group of automorphisms of S and R is the fixed field, then Horn^S, S) is a free S-module with G as basis. Since then, this last condition (“S is G-Galois”) or its equivalents have been used as a postulate in all the Galois theories of rings that are not fields, for example by Dieudonné, Jacobson, Azumaya and Nakayama for noncommutative rings and then in [AG, Appendix] and [CUR] for commutative rings. When S has no idempotents but 0 and 1, [CHR] proves that the ordinary fundamental theorem of Galois theory holds with no real change from the classical, field case.

Group Actions on Quasi-Baer Rings

2009 ◽  
Vol 52 (4) ◽  
pp. 564-582 ◽  
Author(s):  
Hai Lan Jin ◽  
Jaekyung Doh ◽  
Jae Keol Park
Keyword(s):  
Finite Group ◽  
Group Action ◽  
Semiprime Ring ◽  
Group Actions ◽  
Group Rings ◽  
Finite Group Action ◽  
Baer Rings ◽  
Skew Group Rings ◽  
A Finite Group ◽  
The Right

AbstractA ring R is called quasi-Baer if the right annihilator of every right ideal of R is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to C*-algebras. Various examples to illustrate and delimit our results are provided.

scholarly journals The Tracial Class Property for Crossed Products by Finite Group Actions

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Xinbing Yang ◽  
Xiaochun Fang
Keyword(s):  
Finite Group ◽  
Group Actions ◽  
Crossed Products ◽  
Finite Group Actions ◽  
Infinite Dimensional ◽  
A Finite Group ◽  
Tracial Rokhlin Property

We define the concept of tracial𝒞-algebra ofC*-algebras, which generalize the concept of local𝒞-algebra ofC*-algebras given by H. Osaka and N. C. Phillips. Let𝒞be any class of separable unitalC*-algebras. LetAbe an infinite dimensional simple unital tracial𝒞-algebra with the (SP)-property, and letα:G→Aut(A)be an action of a finite groupGonAwhich has the tracial Rokhlin property. ThenA  ×α  Gis a simple unital tracial𝒞-algebra.

scholarly journals The Jordan-Hölder theorem and prefrattini subgroups of finite groups

1995 ◽  
Vol 37 (3) ◽  
pp. 265-277 ◽  
Author(s):  
A. Ballester-Bolinches ◽  
L. M. Ezquerro
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Chief Factors ◽  
One To One ◽  
A Finite Group

All groups considered are finite. In recent years a number of generalizations of the classic Jordan-Hölder Theorem have been obtained (see [7], Theorem A.9.13): in a finite group G a one-to-one correspondence as in the Jordan-Holder Theorem can be defined preserving not only G-isomorphic chief factors but even their property of being Frattini or non-Frattini chief factors. In [2] and [13] a new direction of generalization is presented: the above correspondence can be defined in such a way that the corresponding non-Frattini chief factors have the same complement (supplement).

Extending finite group actions on surfaces to hyperbolic 3-manifolds

1995 ◽  
Vol 117 (1) ◽  
pp. 137-151 ◽  
Author(s):  
Monique Gradolato ◽  
Bruno Zimmermann
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Finite Groups ◽  
Group Actions ◽  
Geodesic Boundary ◽  
Totally Geodesic ◽  
Finite Group Actions ◽  
Closed Riemann Surface ◽  
A Finite Group ◽  
The Given

Let G be a finite group of orientation preserving isometrics of a closed orientable hyperbolic 2-manifold Fg of genus g > 1 (or equivalently, a finite group of conformal automorphisms of a closed Riemann surface). We say that the G-action on Fgbounds a hyperbolic 3-manifold M if M is a compact orientable hyperbolic 3-manifold with totally geodesic boundary Fg (as the only boundary component) such that the G-action on Fg extends to a G-action on M by isometrics. Symmetrically we will also say that the 3-manifold M bounds the given G-action. We are especially interested in Hurwitz actions, i.e. finite group actions on surfaces of maximal possible order 84(g — 1); the corresponding finite groups are called Hurwitz groups. First examples of bounding and non-bounding Hurwitz actions were given in [16].

scholarly journals Conjugacy of free finite group actions on infranilmanifolds

1990 ◽  
Vol 32 (2) ◽  
pp. 239-240 ◽  
Author(s):  
Michał Sadowski
Keyword(s):  
Finite Group ◽  
Group Actions ◽  
Finite Group Actions ◽  
Affine Action ◽  
A Finite Group ◽  
Free Actions

In this note we give the proof of the following result (previously known for homotopically trivial and free actions on infranilmanifolds [3, Theorem 5.6]).Theorem 1. Let G be a finite group acting freely and smoothly on a closed infranilmanifold M. Assume that dim M≠3, 4. Then the action of G is topologically conjugate to an affine action.

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.

Goldie Dimension and Chain Conditions for Modular Lattices with Finite Group Actions

1986 ◽  
Vol 29 (3) ◽  
pp. 274-280 ◽  
Author(s):  
Piotr Grzeszczuk ◽  
Edmund R. Puczyłowski
Keyword(s):  
Finite Group ◽  
Fixed Points ◽  
Modular Lattice ◽  
Group Actions ◽  
Modular Lattices ◽  
Finite Group Actions ◽  
Chain Conditions ◽  
Goldie Dimension ◽  
A Finite Group

AbstractA relation between Goldie dimensions of a modular lattice L and its sublattice LG of fixed points under a finite group G of automorphisms of L is obtained. The method used also gives a relation between ACC (DCC) for L and for LG. The results obtained are applied to rings and modules.

Sign in / Sign up

Export Citation Format

Share Document