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Outer Partial Actions and Partial Skew Group Rings

Published online by Cambridge University Press:  20 November 2018

Patrik Nystedt
Affiliation:
University West, Department of Engineering Science, SE-46186 Trollhättan, Sweden. e-mail: [email protected]
Johan Öinert
Affiliation:
Centre for Mathematical Sciences, P.O. Box 118, Lund University, SE-22100 Lund, Sweden. e-mail: [email protected]
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Abstract

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We extend the classical notion of an outer action $\alpha $ of a group $G$ on a unital ring $A$ to the case when $\alpha $ is a partial action on ideals, all of which have local units. We show that if $\alpha $ is an outer partial action of an abelian group $G$, then its associated partial skew group ring $A\,{{\star }_{\alpha }}\,G$ is simple if and only if $A$ is $G$-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Ánh, P. N. and Márki, L., Morita equivalence for rings without identity. Tsukuba J. Math. 11(1987),no. 1,1–16.Google Scholar
[2] Ávila, J. and Ferrero, M., Closed and prime ideals in partial skew group rings ofabelian groups. J. Algebra Appl. 10(2011), no. 5, 961–978.http://dx.doi.Org/10.1142/S0219498811005063 Google Scholar
[3] Beuter, V. and Gonçalves, D., Partial crossed products as equivalence relation algebras. Rocky Mountain J. Math., to appear. arxiv:1 306.3840Google Scholar
[4] Boava, G. and Exel, R., Partial crossed product description of the C*-algebras associated with integral domains. Proc. Amer. Math. Soc. 141(2013), no. 7, 2439–2451.http://dx.doi.Org/10.1090/S0002-9939-2013-11724-7 Google Scholar
[5] Crow, K., Simple regular skew group rings. J. Algebra Appl. 4(2005), no. 2,127–137.http://dx.doi.org/10.1142/S0219498805000909 Google Scholar
[6] Dokuchaev, M. and Exel, R., Associativity of crossed products by partial actions, enveloping actions and partial representations. Trans. Amer. Math. Soc. 357(2005), no. 5,1931–1952.http://dx.doi.Org/10.1090/S0002-9947-04-0351 9-6 Google Scholar
[7] Dokuchaev, M., Exel, R., and Simon, J. J., Crossed products by twisted partial actions and graded algebras. J. Algebra 320(2008), no. 8, 3278–3310.http://dx.doi.Org/10.1016/j.jalgebra.2008.06.023 Google Scholar
[8] Dokuchaev, M., Del Rio, A., and Simón, J. J., Globalizations of partial actions on nonunital rings. Proc. Amer. Math. Soc. 135(2007), no. 2, 343–352.http://dx.doi.Org/10.1090/S0002-9939-06-08503-0 Google Scholar
[9] Exel, R., Circle actions on C*-algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence. J. Funct. Anal. 122(1994), no. 2, 361–401.http://dx.doi.Org/10.1006/jfan.1994.1073 Google Scholar
[10] Exel, R., The Bunce-Deddens algebras as crossed products by partial automorphisms. Bol. Soc. Brasil. Mat. (N.S.) 25(1994), no. 2,173–179.http://dx.doi.Org/10.1007/BF01321306 Google Scholar
[11] Exel, R., Approximately finite C*-algebras and partial automorphisms. Math. Scand. 77(1995),no. 2, 281–288.Google Scholar
[12] Exel, R., Partial actions of groups and actions of inverse semigroups. Proc. Amer. Math. Soc. 126(1998), no. 12, 3481–3494.http://dx.doi.Org/10.1090/S0002-9939-98-04575-4 Google Scholar
[13] Exel, R., Giordano, T., and Goncalves, D., Enveloping algebras of partial actions as groupoid C*-algebras. J. Operator Theory 65(2011), no. 1, 197–210.Google Scholar
[14] Exel, R. and Laca, M., Cuntz-Krieger algebras for infinite matrices. J. Reine Angew. Math. 512(1999), 119–172. http://dx.doi.Org/10.1515/crll.1999.051Google Scholar
[15] Fisher, J. W. and Montgomery, S., Semiprime skew group rings. J. Algebra 52(1978), no. 1, 241–247.http://dx.doi.Org/10.1016/0021-8693(78)90272-7 Google Scholar
[16] Gonçalves, D., Simplicity of partial skew group rings of abelian groups. Canad. Math. Bull. 57(2014), no. 3, 511–519. http://dx.doi.Org/10.4153/CMB-2014-011-1Google Scholar
[17] Gonçalves, D., Öinert, J., and Royer, D., Simplicity of partial skew group rings with applications to Leavittpath algebras and topological dynamics. J. Algebra 420(2014), 201–216.http://dx.doi.Org/10.1016/j.jalgebra.2O14.07.027 Google Scholar
[18] Haefner, J. and del Rio, A., The globalization problem for inner automorphisms and Skolem-Noether theorems. In: Algebras, rings and their representations, World Sci. Publ., Hackensack, NJ, 2006, pp. 37–51. http://dx.doi.Org/10.1142/9789812774552.0005 Google Scholar
[19] Lam, T. Y., A first course in noncommutative rings. Springer-Verlag, New York, 1991.http://dx.doi.Org/10.1007/978-1-4684-0406-7 Google Scholar
[20] McClanahan, K., K-theory for partial crossed products by discrete groups. J. Funct. Anal. 130(1995),no. 1, 77–117.http://dx.doi.Org/10.1006/jfan.1995.1064 Google Scholar
[21] Nystedt, P. and Öinert, J., Simple semigroup graded rings. J. Algebra Appl., to appear.http://dx.doi.Org/10.1142/S0219498815501029 Google Scholar
[22] Öinert, J., Simplicity of skew group rings ofabelian groups. Comm. Algebra 42(2014), no. 2,831–841.http://dx.doi.org/10.1080/00927872.2012.727052 Google Scholar