Infinite crossed products and group-graded rings

Passman, D. S.

doi:10.1090/s0002-9947-1984-0743740-2

Cited by 20 publications

(28 citation statements)

References 22 publications

(21 reference statements)

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“…Passman's example of a field which is a twisted group algebra of an infinite periodic group ( [15,Proposition 4.3]) shows that the restriction on periodic subgroups cannot be removed from Corollaries 10 and 11.…”

Section: Corollary 10 Let S Be a Linear Cancellative Semigroup With mentioning

confidence: 99%

Radicals of algebras graded by cancellative linear semigroups

Kelarev¹

1996

Proc. Amer. Math. Soc.

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Abstract. We consider algebras over a field of characteristic zero, and prove that the Jacobson radical is homogeneous in every algebra graded by a linear cancellative semigroup. It follows that the semigroup algebra of every linear cancellative semigroup is semisimple.Recently it has been shown that several theorems concerning various ring constructions can be obtained in the more general situation of graded rings. One of the long-standing problems on group algebras is that of whether every group algebra over any field of characteristic zero is semiprimitive. The answer is known to be positive for linear groups. We obtain a graded analog of this result.Group algebras of linear groups have been explored by many authors. Passman and Zalesskii investigated the Jacobson radical and semiprimitivity of these algebras, see [16]. A systematic study of the semigroup algebras of linear semigroups has been started by Okniński and Putcha ([12], [13], [14]). In particular, in [14] the radicals of algebras of connected algebraic monoids were described. For algebras over a field of characteristic zero, it is shown in [13, §3] that the radical of every algebra of a linear semigroup is nilpotent, and the algebra of the full matrix semigroup is semiprimitive. For the full matrix semigroup M over a finite field F and for a field K of characteristic different from that of F , it follows from Fadeev's Theorem (see Kovács [10]) that the semigroup algebra KM is semiprimitive.Let S be a semigroup. An algebra R = s∈S R s is said to be S-graded if R s R t ⊆ R st for all s, t ∈ S. An ideal I of R is said to be homogeneous if I = s∈S I s , where I s = I ∩ R s . Theorem 1. Let S be a cancellative linear semigroup. Then the Jacobson radical of every S-graded algebra over a field of characteristic zero is homogeneous.The radical of a (not necessarily cancellative) semigroup algebra over a field cannot contain nonzero homogeneous elements. Indeed, the factor of the semigroup

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Section: Corollary 10 Let S Be a Linear Cancellative Semigroup With mentioning

confidence: 99%

Radicals of algebras graded by cancellative linear semigroups

Kelarev¹

1996

Proc. Amer. Math. Soc.

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“…If in addition, S x S y = S xy holds for all x, y ∈ G, then S is said to be strongly G-graded. An interesting problem, studied for the past 50 years, concerns finding necessary and sufficient conditions for different classes of group graded rings to be prime, see [4,9,10,28,29,35,36,37,38,39,40]. In the case when S is unital and strongly G-graded, Passman has completely solved this problem by proving the following rather involved result: Theorem 1.1 (Passman [40,Thm.…”

Section: Introductionmentioning

confidence: 99%

“…If x ∈ G, then I x denotes the S e -ideal S x −1 IS x . Let H, N be subgroups of G. The ideal I is called H-invariant if I x ⊆ I holds for every x ∈ H; S N denotes x∈N S x , which is clearly a subring of S. In [40] Passman provided a "combinatorial" proof of Theorem 1.1 by combining two main ideas. First, a coset counting method, also known as the "∆-method", developed by Passman [35] and Connell [10], secondly, the "bookkeeping procedure" introduced by Passman in [38] which involves a careful study of the action of the group G on the lattice of ideals of S e .…”

Section: Introductionmentioning

confidence: 99%

Prime group graded rings with applications to partial crossed products and Leavitt path algebras

Lännström,

Lundström,

Öinert

et al. 2021

Preprint

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In this article we generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a characterization of prime s-unital strongly group graded rings, and, in particular, of infinite matrix rings and of group rings over s-unital rings, thereby generalizing a well-known result by Connell; (ii) characterizations of prime s-unital partial skew group rings and of prime unital partial crossed products; (iii) a generalization of the well-known characterizations of prime Leavitt path algebras, by Larki and by Abrams-Bell-Rangaswamy.

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“…In a crossed product D * A (see [19,Ch. 1]), where D is a division ring, the multiplication is defined as in (ii) above, but an element d ∈ D need not be central.…”

Section: Introductionmentioning

confidence: 99%

“…It is known (see, for example, [19,Lemma 37.8]) that if B is a subgroup of A, then S B = F * B \ {0} is an Ore subset in F * A. As a consequence, the subset T S B (M) of M, consisting of all x such that xs = 0 for some s ∈ S B , is an F * A-submodule of M. We say that M is S B -torsion or F * B-torsion if T S B (M) = M and S B -torsion-free or F * B-torsion-free if T S B (M) = 0.…”

Section: Introductionmentioning

confidence: 99%

Modules Over Quantum Laurent Polynomials

Gupta¹

2011

J. Aust. Math. Soc.

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We show that the Gelfand-Kirillov dimension for modules over quantum Laurent polynomials is additive with respect to tensor products over the base field. We determine the Brookes-Groves invariant associated with a tensor product of modules. We study strongly holonomic modules and show that there are nonholonomic simple modules.2010 Mathematics subject classification: primary 16S35; secondary 16D30, 16D60.

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Infinite crossed products and group-graded rings

Cited by 20 publications

References 22 publications

Radicals of algebras graded by cancellative linear semigroups

Radicals of algebras graded by cancellative linear semigroups

Prime group graded rings with applications to partial crossed products and Leavitt path algebras

Modules Over Quantum Laurent Polynomials

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