Semisimple strongly graded rings
type: Item
instance of: scholarly article
title: Semisimple strongly graded rings
zbMATH Open document ID: 1019.16029
DOI: 10.1016/S0021-8693(02)00113-8
author: Eli Aljadeff
author: Yuval Ginosar
author: Ángel del Río
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: Let \(G\) be a finite group with neutral element \(1\). A strongly \(G\)-graded ring \(R=\bigoplus_{g\in G}R_g\) induces a group homomorphism \(\sigma\colon G\to\text{Pic}(A)\), where \(A=R_1\), and \(\text{Pic}(A)\) denotes the Picard group of the ring \(A\). When \(R\) is a crossed product \(R=A*^\beta_tG\), then this homomorphism comes from the action \(\beta\colon G\to\Aut(A)\) via the canonical map \(\Aut(A)\to\text{Pic}(A)\). Recall that in such a case, the product of \(R\) is defined from a right \(A\)-basis of homogeneous units \(\{u_g:g\in G\}\), namely \(au_g=u_ga^{\beta(g)}\), \(u_gu_h=u_{hg}t(g,h)\), where \(t\colon G\times G\to A^*\) is the twisting. If \(A\) is a division ring with center \(K\), and \(H\) is the kernel of the outer action \(\alpha\colon G\to\text{Out}(A)\) induced by \(\beta\), then \(A*_{t'}H\) is a twisted group ring (i.e. with trivial action), where \(t'\) is a twisting obtained after a diagonal change of basis. Moreover, \(K*_{t'}H\) makes now sense, as \(t'(g,h)\in K\) for every \(g,h\in H\). It is proved (Theorem 3.2) that \(R=A*^\beta_tG\) is semisimple if and only if \(A*_{t'}H\) is semisimple if and only if \(K*_{t'}H\) is semisimple. This leads, in conjunction with Morita theory and the work by \textit{E. Aljadeff} and \textit{D. J. S. Robinson} [J. Pure Appl. Algebra 94, No. 1, 1-15 (1994; Zbl 0809.16029)], to several interesting characterizations of strongly \(G\)-graded semisimple rings (Theorem 3.3). As a consequence, the group homomorphism \(\sigma\colon G\to\text{Pic}(A)\) satisfies some necessary conditions to be induced by a strongly graded semisimple ring. The authors study whether these conditions are also sufficient (this is the `twisting problem'), and give a positive answer when \(G\) is cyclic and \(A\) is finitely generated and projective over its center (Theorem 4.6). The paper is completed with an application to actions of finite groups on division rings of prime characteristic (Theorem 5.1). If \(H\) is the …
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16S35
zbMATH DE Number: 1868024
zbMATH Keywords: strongly group-graded rings
zbMATH Keywords: crossed products
zbMATH Keywords: semisimple rings
zbMATH Keywords: Galois actions
zbMATH Keywords: Picard groups
reviewed by: José Gómez-Torrecillas
MaRDI profile type: Publication
cites work: Q4847007
cites work: Projective representations and relative semisimplicity
cites work: Semisimple algebras, Galois actions and group cohomology
cites work: Solution of the twisting problem for skew group algebras
cites work: Q5562621
cites work: Obstructions to Clifford system extensions of algebras
cites work: Produits croisés mixtes: extensions de groupes et extensions d'anneaux
cites work: Actions of Picard groups on graded rings
cites work: Q4339908
cites work: Descending chain conditions and graded rings
cites work: When is R-gr equivalent to the category of modules?
cites work: Fixed rings of finite automorphism groups of associative rings
cites work: Q3957087
cites work: Separable functors applied to graded rings
cites work: Infinite Crossed Products and Group-Graded Rings
cites work: Q4040049
cites work: Q3856855
cites work: The finite inner automorphism groups of division rings
cites work: Q5587110
full work available at URL: s0021-8693(02)00113-8
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description: scientific article; zbMATH DE number 1868024