scholarly journals The twisted group ring isomorphism problem over fields

2020 ◽  
Vol 238 (1) ◽  
pp. 209-242
Author(s):  
Leo Margolis ◽  
Ofir Schnabel
Keyword(s):  
Group Ring ◽  
Isomorphism Problem ◽  
Ring Isomorphism ◽  
Twisted Group ◽  
Twisted Group Ring

Schur and Projective Schur Groups of Number Rings

1991 ◽  
Vol 43 (3) ◽  
pp. 540-558 ◽  
Author(s):  
Peter Nelis
Keyword(s):  
Group Ring ◽  
Group Algebra ◽  
Number Fields ◽  
Commutative Rings ◽  
Azumaya Algebras ◽  
Twisted Group Algebra ◽  
Twisted Group ◽  
Simple Algebras ◽  
Central Simple ◽  
Twisted Group Ring

The Schur or projective Schur group of a field consists of the classes of central simple algebras which occur in the decomposition of a group algebra or a twisted group algebra. For number fields, the projective Schur group has been determined in [8], whereas the Schur group is extensively studied in [25]. Recently, some authors have generalized these concepts to commutative rings. One then studies the classes of Azumaya algebras which are epimorphic images of a group ring or a twisted group ring. Though several properties of the Schur or projective Schur group defined in this way have been obtained, they remain rather obscure objects.

scholarly journals Twisted group rings which are semi-prime Goldie rings

1975 ◽  
Vol 16 (1) ◽  
pp. 1-11 ◽  
Author(s):  
A. Reid
Keyword(s):  
Group Ring ◽  
Quotient Ring ◽  
Factor Group ◽  
Polynomial Rings ◽  
Group Rings ◽  
Twisted Group ◽  
Twisted Group Ring ◽  
Quotient Rings

In this paper we examine when a twisted group ring,Rγ(G), has a semi-simple, artinian quotient ring. In §1 we assemble results and definitions concerning quotient rings, Ore sets and Goldie rings and then, in §2, we defineRγ(G). We prove a useful theorem for constructing a twisted group ring of a factor group and establish an analogue of a theorem of Passman. Twisted polynomial rings are discussed in §3 and I am indebted to the referee for informing me of the existence of [4]. These are used as a tool in proving results in §4.

scholarly journals Twisted group rings and a problem of Faith

1973 ◽  
Vol 9 (1) ◽  
pp. 11-19 ◽  
Author(s):  
John H. Cozzens
Keyword(s):  
Group Ring ◽  
Type Theorem ◽  
Arbitrary Field ◽  
Group Rings ◽  
One Dimensional ◽  
Simple Modules ◽  
Isomorphism Classes ◽  
Twisted Group ◽  
Homological Characterization ◽  
Twisted Group Ring

A homological characterization is given of when a twisted group ring relative to an automorphism of an arbitrary field has all of its simple right modules injective (= a right V-ring). This answers a question raised by Osofsky. A “Hilbert Theorem 90” type theorem determines the cardinality of the isomorphism classes of one-dimensional simple modules.

scholarly journals Integral representations of a certain twisted group ring

1989 ◽  
Vol 124 (1) ◽  
pp. 1-8 ◽  
Author(s):  
A Chalatsis ◽  
Th Theohari-Apostolidi
Keyword(s):  
Group Ring ◽  
Integral Representations ◽  
Twisted Group ◽  
Twisted Group Ring

On Automorphisms of Complete Algebras And The Isomorphism Problem for Modular Group Rings

1990 ◽  
Vol 42 (3) ◽  
pp. 383-394 ◽  
Author(s):  
Frank Röhl
Keyword(s):  
Group Ring ◽  
Modular Group ◽  
Isomorphism Problem ◽  
Nilpotency Class ◽  
Affirmative Answer ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Analogous Question

In [5], Roggenkamp and Scott gave an affirmative answer to the isomorphism problem for integral group rings of finite p-groups G and H, i.e. to the question whether ZG ⥲ ZH implies G ⥲ H (in this case, G is said to be characterized by its integral group ring). Progress on the analogous question with Z replaced by the field Fp of p elements has been very little during the last couple of years; and the most far reaching result in this area in a certain sense - due to Passi and Sehgal, see [8] - may be compared to the integral case, where the group G is of nilpotency class 2.

scholarly journals Public Key Protocols over Twisted Dihedral Group Rings

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1019
Author(s):  
María Dolores Gómez Olvera ◽  
Juan Antonio López Ramos ◽  
Blas Torrecillas Jover
Keyword(s):  
Information Security ◽  
Group Ring ◽  
Key Management ◽  
Dihedral Group ◽  
Complexity Analysis ◽  
Public Key ◽  
Group Rings ◽  
Decomposition Problem ◽  
Hard Problems ◽  
Twisted Group Ring

Key management is a central problem in information security. The development of quantum computation could make the protocols we currently use unsecure. Because of that, new structures and hard problems are being proposed. In this work, we give a proposal for a key exchange in the context of NIST recommendations. Our protocol has a twisted group ring as setting, jointly with the so-called decomposition problem, and we provide a security and complexity analysis of the protocol. A computationally equivalent cryptosystem is also proposed.

scholarly journals The rank of syzygies under the action by a finite group

1978 ◽  
Vol 71 ◽  
pp. 1-12 ◽  
Author(s):  
Shiro Goto
Keyword(s):  
Finite Group ◽  
Local Ring ◽  
Group Ring ◽  
Group Algebra ◽  
Maximal Ideal ◽  
Residue Field ◽  
Noetherian Local Ring ◽  
Trace Ideal ◽  
A Finite Group ◽  
Twisted Group Ring

Let S be a Noetherian local ring with maximal ideal J and k the residue field of S. Let G be a finite group of order n and suppose that G acts on S as automorphisms. Let R = SG and I = JG. We denote by S[G] (resp. R[G]) the twisted group ring of G over S (resp. the group algebra of G over R). Recall that the multiplication of S[G] is defined as follows : sg · th = sg(t) · gh for s, t ∊ S and g, h ∊ G. Let tG(S) = {Σg ∈ Gg(s)/s ∈ S} and call it the trace ideal of S. Note that tG(S) = R if n is a unit of S. We say that S has a normal basis if S ≅ R[G] as R[G]-modules. This condition says that there is an element s of S so that {g(s)}g ∈ G forms an R-free basis of S.

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