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George Szeto - One of the best experts on this subject based on the ideXlab platform.
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On Galois Extensions for Separable Group Rings
2020Co-Authors: George SzetoAbstract:Let R be a ring with 1, G a group, and RG a group ring with center C. Assume RG is an Azumaya C-algebra. Then the inner automorphism group G of RG induced by the elements of G is finite, and RG is not a Galois Extension of (RG) G with Galois group G. For a proper subgroup K of G with an invertible order, the following are equivalent: (1) RG is a Galois Extension of (RG) K with Galois group K; (2) RG is a projective right (RG) K -module and the centralizer of (RG) K is ⊕ g∈K Jg where Jg = {a ∈ RG ax = g(x)a for each x ∈ RG}; and (3) {g ∈ G g is a representative of g ∈ K} are linearly independent over C. Moreover, we call f : K −→ (RG) K the Galois map from the set of subgroups of G to the set of subalgebras of RG. Then f is one-to-one from a set of Galois groups K of RG to the set of separable subalgebras (RG) K of RG. Mathematics Subject Classification: 16S35, 16W20
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A Structure of Galois Extensions with an Inner Galois Group
2020Co-Authors: George SzetoAbstract:Let B be a Galois Extension of B G with an inner Galois group G where G = {gi |gi(x )= UixU −1 i for some Ui ∈ B and for all x ∈ B, i =1 ,2, ··· ,nfor some integer n invertible in B}. Then B is a composition of two Galois Extensions B ⊃ B K with an inner Abelian Galois group K and B K ⊃ B G with an inner Galois group G/K where K = {g ∈ G |g(Ui )= Ui for each i}. Descriptions of B ⊃ B K and B K ⊃ B G are given.
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Galois Extensions with a Galois Commutator Subring
2020Co-Authors: George SzetoAbstract:Let B be a Galois Extension of B G with Galois group G, Δ the commutator subring of B G in B, and G|Δ the restriction of G to Δ. Equivalent conditions are given for a Galois Extension Δ of Δ G with Galois group G|Δ. It is shown that the following statements are equivalent: (1) Δ is a Galois Extension of Δ G with Galois group induced by and isomorphic with G/N where N = {g ∈ G |g(x )= x for all x ∈ Δ}. (2) B G Δ is a Galois Extension of B G with Galois group induced by and isomorphic with G/N and Δ is a finitely generated and projective module over Δ G . (3) B is a composition of two Galois Extensions: B ⊃ B G Δ with Galois group N and B G Δ ⊃ B G with Galois group induced by and isomorphic with G/N such that Δ is a finitely generated and projective module over Δ G . Consequently, more results can be derived for several well known classes of Galois Extensions such as DeMeyer-Kanzaki Galois Extensions, Azumaya Galois Extensions, and Hirata separable Galois Extensions.
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On three special types of partial Galois Extensions
Journal of Algebra and Its Applications, 2019Co-Authors: Xiaolong Jiang, George SzetoAbstract: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 exte...
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Partial group actions and partial Galois Extensions
Monatshefte für Mathematik, 2016Co-Authors: George SzetoAbstract: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.
Hai Q Dinh - One of the best experts on this subject based on the ideXlab platform.
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constacyclic codes of length over Galois Extension rings of
IEEE Transactions on Information Theory, 2009Co-Authors: Hai Q DinhAbstract:We study all constacyclic codes of length over GR , the Galois Extension ring of dimension of the ring . The units of the ring GR are of the forms , and , where are nonzero elements of , which correspond to such constacyclic codes. First, the structure and Hamming distances of -constacyclic codes are established. We then classify all cyclic codes of length over GR , and obtain a formula for the number of those cyclic codes, as well as the number of codewords in each code. Finally, one-to-one correspondences between cyclic and -constacyclic codes, as well as -constacyclic and -constacyclic codes are provided via ring isomorphisms, that allow us to carry over the results about cyclic and -constacyclic accordingly to all constacyclic codes of length over GR .
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constacyclic codes of length 2 s over Galois Extension rings of bbf _ 2 u bbf _2
IEEE Transactions on Information Theory, 2009Co-Authors: Hai Q DinhAbstract:We study all constacyclic codes of length 2s over GR(Rfr,m), the Galois Extension ring of dimension m of the ring Rfr=F2+uF2. The units of the ring GR(Rfr,m) are of the forms alpha, and alpha+ubeta, where alpha, beta are nonzero elements of F2m, which correspond to 2 m(2m-1) such constacyclic codes. First, the structure and Hamming distances of (1+ugamma)-constacyclic codes are established. We then classify all cyclic codes of length 2s over GR(Rfr,m), and obtain a formula for the number of those cyclic codes, as well as the number of codewords in each code. Finally, one-to-one correspondences between cyclic and alpha-constacyclic codes, as well as (1+ugamma)-constacyclic and (alpha+ubeta) -constacyclic codes are provided via ring isomorphisms, that allow us to carry over the results about cyclic and (1+ugamma)-constacyclic accordingly to all constacyclic codes of length 2s over GR(Rfr,m).
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Constacyclic Codes of Length $2^s$ Over Galois Extension Rings of ${\BBF}_{2}+u{\BBF}_2$
IEEE Transactions on Information Theory, 2009Co-Authors: Hai Q DinhAbstract:We study all constacyclic codes of length 2s over GR(Rfr,m), the Galois Extension ring of dimension m of the ring Rfr=F2+uF2. The units of the ring GR(Rfr,m) are of the forms alpha, and alpha+ubeta, where alpha, beta are nonzero elements of F2m, which correspond to 2 m(2m-1) such constacyclic codes. First, the structure and Hamming distances of (1+ugamma)-constacyclic codes are established. We then classify all cyclic codes of length 2s over GR(Rfr,m), and obtain a formula for the number of those cyclic codes, as well as the number of codewords in each code. Finally, one-to-one correspondences between cyclic and alpha-constacyclic codes, as well as (1+ugamma)-constacyclic and (alpha+ubeta) -constacyclic codes are provided via ring isomorphisms, that allow us to carry over the results about cyclic and (1+ugamma)-constacyclic accordingly to all constacyclic codes of length 2s over GR(Rfr,m).
Joachim König - One of the best experts on this subject based on the ideXlab platform.
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The Grunwald problem and specialization of families of regular Galois Extensions
arXiv: Number Theory, 2017Co-Authors: Joachim KönigAbstract:We investigate specializations of infinite families of regular Galois Extensions over number fields. The problem to what extent the local behaviour of specializations of one single regular Galois Extension can be prescribed has been investigated by D\`ebes and Ghazi in the unramified case, and by Legrand, Neftin and the author in general. Here, we generalize these results and give a partial solution to Grunwald problems using Galois Extensions arising as specializations of a family of regular Galois Extensions.
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Non-parametricity of rational translates of regular Galois Extensions
arXiv: Number Theory, 2016Co-Authors: Joachim KönigAbstract:We generalize a result of F.\ Legrand about the existence of non-parametric Galois Extensions for a given group $G$. More precisely, for a $K$-regular Galois Extension $F|K(t)$, we consider the translates $F(s)|K(s)$ by an Extension $K(s)|K(t)$ of rational function fields (in other words, $s$ is a root of $g(X)-t$ for some rational function $g\in K(X)$). We then show that if $F|K(t)$ is a $K$-regular Galois Extension with group $G$ over a number field $K$, then for any degree $k\ge 2$ and almost all (in a density sense) rational functions $g$ of degree $k$, the translate of $F$ by a root field of $g(X)-t$ over $K(t)$ is non-$G$-parametric, i.e.\ not all Galois Extensions of $K$ with group $G$ arise as specializations of $F(s)|K(s)$.
Aiping Zhang - One of the best experts on this subject based on the ideXlab platform.
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finitistic dimension of faithfully flat weak hopf Galois Extension
arXiv: Representation Theory, 2018Co-Authors: Aiping ZhangAbstract:Let $H$ be a finite-dimensional weak Hopf algebra over a field $k$ and $A/B$ be a right faithfully flat weak $H$-Galois Extension. We prove that if the finitistic dimension of $B$ is finite, then it is less than or equal to that of $A$. Moreover, suppose that $H$ is semisimple. If the finitistic dimension conjecture holds, then the finitistic dimension of $B$ is equal to that of $A$.
Xiaolong Jiang - One of the best experts on this subject based on the ideXlab platform.
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On three special types of partial Galois Extensions
Journal of Algebra and Its Applications, 2019Co-Authors: Xiaolong Jiang, George SzetoAbstract: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 exte...
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Galois Extensions induced by a central idempotent in a partial Galois Extension
International Journal of Algebra, 2014Co-Authors: Xiaolong Jiang, George SzetoAbstract: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
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the Galois endomorphism ring of a Galois azumaya Extension
International Journal of Algebra, 2013Co-Authors: Xiaolong Jiang, George SzetoAbstract:Let B be a Galois Azumaya Extension of B G with Galois group G; that is, B is a Galois Extension of B G with Galois group G which is an Azumaya C G -algebra where C is the center of B. Denote B G by D and the endomorphism ring Hom(DB, DB) of the left D-module endomorphisms of B by Ω. Then Ω is a Galois and a Hirata separable
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on composition series of a general azumaya Galois Extension
2012Co-Authors: Xiaolong Jiang, George SzetoAbstract:Let B be a ring with 1 and a Galois Extension of B G with Galois group G such that B G is a separable C G −algebra where C is the center of B, H a subgroup of G andH the commutator subring of B H in B. IfH is a Galois algebra with Galois group N/N0 where N is the normalizer of H in G and N0 = {g ∈ N, g(x) = x for all x ∈ �H }, then we show two compositions of Galois Extensions, one for B over B N , and another forH over (�H) N.
