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Search: MSC category 16S35 ( Twisted and skew group rings, crossed products )

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1. CMB 2014 (vol 57 pp. 511)

Gonçalves, Daniel
Simplicity of Partial Skew Group Rings of Abelian Groups
Let $A$ be a ring with local units, $E$ a set of local units for $A$, $G$ an abelian group and $\alpha$ a partial action of $G$ by ideals of $A$ that contain local units. We show that $A\star_{\alpha} G$ is simple if and only if $A$ is $G$-simple and the center of the corner $e\delta_0 (A\star_{\alpha} G) e \delta_0$ is a field for all $e\in E$. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.

Keywords:partial skew group rings, simple rings, partial actions, abelian groups
Categories:16S35, 37B05

2. CMB 2009 (vol 53 pp. 321)

Lee, Tsiu-Kwen; Zhou, Yiqiang
A Theorem on Unit-Regular Rings
Let $R$ be a unit-regular ring and let $\sigma $ be an endomorphism of $R$ such that $\sigma (e)=e$ for all $e^2=e\in R$ and let $n\ge 0$. It is proved that every element of $R[x \mathinner;\sigma]/(x^{n+1})$ is equivalent to an element of the form $e_0+e_1x+\dots +e_nx^n$, where the $e_i$ are orthogonal idempotents of $R$. As an application, it is proved that $R[x \mathinner; \sigma ]/(x^{n+1})$ is left morphic for each $n\ge 0$.

Keywords:morphic rings, unit-regular rings, skew polynomial rings
Categories:16E50, 16U99, 16S70, 16S35

3. CMB 2009 (vol 52 pp. 564)

Jin, Hai Lan; Doh, Jaekyung; Park, Jae Keol
Group Actions on Quasi-Baer Rings
A ring $R$ is called {\it quasi-Baer} if the right annihilator of every right ideal of $R$ is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to $C^*$-algebras. Various examples to illustrate and delimit our results are provided.

Keywords:(quasi-) Baer ring, fixed ring, skew group ring, $C^*$-algebra, local multiplier algebra
Categories:16S35, 16W22, 16S90, 16W20, 16U70

4. CMB 2008 (vol 51 pp. 261)

Neeb, Karl-Hermann
On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups
An $n$-dimensional quantum torus is a twisted group algebra of the group $\Z^n$. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational $n$-dimensional quantum tori over any field. Moreover, we show that for $n = 2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field.

Keywords:quantum torus, normal form, automorphisms of quantum tori
Category:16S35

5. CMB 2002 (vol 45 pp. 388)

Gille, Philippe
Algèbres simples centrales de degré 5 et $E_8$
As a consequence of a theorem of Rost-Springer, we establish that the cyclicity problem for central simple algebra of degree~5 on fields containg a fifth root of unity is equivalent to the study of anisotropic elements of order 5 in the split group of type~$E_8$.

Keywords:algèbres simples centrales, cohomologie galoisienne
Categories:16S35, 12G05, 20G15

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