Twisted group rings and a problem of Faith

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.

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Twisted group rings which are semi-prime Goldie rings

Glasgow Mathematical Journal ◽

10.1017/s0017089500002433 ◽

1975 ◽

Vol 16 (1) ◽

pp. 1-11 ◽

Cited By ~ 6

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.

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q-Analogue of a Twisted Group Ring

Finite Reductive Groups: Related Structures and Representations ◽

10.1007/978-1-4612-4124-9_1 ◽

1997 ◽

pp. 1-13

Author(s):

Susumi Ariki

Keyword(s):

Group Ring ◽

Twisted Group ◽

Twisted Group Ring

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On some properties of group rings

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700021534 ◽

1980 ◽

Vol 29 (4) ◽

pp. 385-392 ◽

Cited By ~ 1

Author(s):

G. Karpilovsky

Keyword(s):

Commutative Ring ◽

Number Field ◽

Group Ring ◽

Algebraic Number ◽

Algebraic Number Field ◽

Group Rings ◽

Arbitrary Group ◽

Bijective Correspondence ◽

Isomorphism Classes ◽

Ring Of Algebraic Integers

AbstractLet Out (RG) be the set of all outer R-automorphisms of a group ring RG of arbitrary group G over a commutative ring R with 1. It is proved that there is a bijective correspondence between the set Out (RG) and a set consisting of R(G × G)-isomorphism classes of R-free R(G × G)-modules of a certain type. For the case when G is finite and R is the ring of algebraic integers of an algebraic number field the above result implies that there are only finitely many conjugacy classes of group bases in RG. A generalization of a result due to R. Sandling is also provided.

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Public Key Protocols over Twisted Dihedral Group Rings

Symmetry ◽

10.3390/sym11081019 ◽

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.

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Schur and Projective Schur Groups of Number Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1991-033-5 ◽

1991 ◽

Vol 43 (3) ◽

pp. 540-558 ◽

Cited By ~ 4

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.

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