Twisted group rings which are semi-prime Goldie 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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Artinian quotient rings of group rings

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700015202 ◽

1973 ◽

Vol 16 (3) ◽

pp. 379-384 ◽

Cited By ~ 6

Author(s):

Ian Hughes

Keyword(s):

Finite Group ◽

Finite Number ◽

Group Ring ◽

Quotient Ring ◽

Group Rings ◽

Normal Series ◽

Quotient Rings

Smith [6, Theorem 2.18] proved that if A is a ring which has a right artinian right quotient ring and G is a poly- (cyclic-or-finite) group, then the group ring AG has a right artinian right quotient ring. We give here a different proof (and a generalization) of this result using methods developed by Jategaonkar [3,4]. THEOREM. Let A be a ring which has a right artinian right quotient ring, and let G be a group which has a (transfinite) ascending normal series with each factor either finite or cyclic, but only a finite number of finite factors. Then AG has a right artinian right quotient ring.

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Twisted group rings and a problem of Faith

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700042817 ◽

1973 ◽

Vol 9 (1) ◽

pp. 11-19 ◽

Cited By ~ 1

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.

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Quasi-Frobenius quotient rings of group rings

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700016128 ◽

1975 ◽

Vol 20 (4) ◽

pp. 394-397 ◽

Cited By ~ 1

Author(s):

Andreas Horn

Keyword(s):

Finite Number ◽

Group Ring ◽

Quotient Ring ◽

Group Rings ◽

Normal Series ◽

Quotient Rings

The purpose of this paper is an extension of a theorem of Hughes (1973). He showed: Let R be a ring which has a right artinian right quotient ring and let G be a group which has a (transfinite) ascending normal series with each factor either finite or cyclic, but only a finite number of finite factors. Then the group ring RG has a right artinian right quotient ring.

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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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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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