Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T10:58:35.680Z Has data issue: false hasContentIssue false
  • English
  • Français

Group Actions on Quasi-Baer Rings

Published online by Cambridge University Press:  20 November 2018

Hai Lan Jin ,
Jaekyung Doh  and
Jae Keol Park
Hai Lan Jin
Affiliation:
Department of Mathematics, Yanbian University, Yanji 133002, People's Republic of China e-mail: [email protected]
Jaekyung Doh
Affiliation:
Department of Mathematics, Busan National University, Busan 609–735, South Korea e-mail: [email protected]
Jae Keol Park
Affiliation:
Department of Mathematics, Busan National University, Busan 609–735, South Korea e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A ring $R$ is called 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

16S3516W2216S9016W2016U70(quasi-) Baer ringfixed ringskew group ringC*-algebralocal multiplier algebra
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 4 , 01 December 2009 , pp. 564 - 582
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Amitsur, S. A., On rings of quotients. In: Symposia Mathematica VIII, Academic Press, London, 1972, pp. 149–164.Google Scholar
[2] Ara, P., The extended centroid of C*-algebras. Arch. Math. 54(1990), no. 4, 358–364.Google Scholar
[3] Ara, P., On the symmetric algebra of quotients of a C*-algebra. Glasgow Math. J. 32(1990), no. 3, 377–379.Google Scholar
[4] Ara, P. and Mathieu, M., A local version of the Dauns-Hofmann theorem. Math. Z. 208(1991), no. 3, 349–353.Google Scholar
[5] Ara, P. and Mathieu, M., An application of local multipliers to centralizing mappings of C*-algebras. Quart. J. Math. Oxford 44(1993), no. 174, 129–138.Google Scholar
[6] Ara, P. and Mathieu, M., Local Multipliers of C*-Algebras. Springer-Verlag, London, 2003.Google Scholar
[7] Armendariz, E. P., A note on extensions of Baer and P.P.-rings. J. Austral. Math. Soc. 18(1974), 470–473.Google Scholar
[8] Bergman, G. M. and Issacs, I. M., Rings with fixed-point-free group actions. Proc. London Math. Soc. 27(1973), 69–87.Google Scholar
[9] Birkenmeier, G. F., Idempotents and completely semiprime ideals. Comm. Algebra 11(1983), no. 6, 567–580.Google Scholar
[10] Birkenmeier, G. F., Decompositions of Baer-like rings. Acta Math. Hungar. 59(1992), no. 3-4, 319–326.Google Scholar
[11] Birkenmeier, G. F., Călugăreanu, G., Fuchs, L., and Goeters, H. P., The fully invariant-extending property for abelian groups. Comm. Algebra 29(2001), no. 2, 673–685.Google Scholar
[12] Birkenmeier, G. F., Heatherly, H. E., Kim, J. Y., and Park, J. K., Triangular matrix representations. J. Algebra 230(2000), no. 2, 558–595.Google Scholar
[13] Birkenmeier, G. F., Kim, J. Y., and Park, J. K., Quasi-Baer ring extensions and biregular rings. Bull. Austral.Math. Soc. 61(2000), no. 1, 39–52.Google Scholar
[14] Birkenmeier, G. F., Kim, J. Y., and Park, J. K., A sheaf representation of quasi-Baer rings. J. Pure Appl. Algebra 146(2000), no. 3, 209–223.Google Scholar
[15] Birkenmeier, G. F., Kim, J. Y., and Park, J. K., On quasi-Baer rings. In: Algebra and its applications, Contemp. Math. 259, American Mathematical Society, Providence, RI, 2000, pp. 67–92.Google Scholar
[16] Birkenmeier, G. F., Kim, J. Y., and Park, J. K., Semicentral reduced algebras. In: International symposiumon ring theory, Birkhäuser, Boston, MA, 2001, pp. 67–84.Google Scholar
[17] Birkenmeier, G. F., Kim, J. Y., and Park, J. K., Polynomial extensions of Baer and quasi-Baer rings. J. Pure Appl. Algebra 159(2001), no. 1, 25–42.Google Scholar
[18] Birkenmeier, G. F., Müller, B. J., and Rizvi, S. T., Modules in which every fully invariant submodule is essential in a direct summand. Comm. Algebra 30(2002), no. 3, 1395–1415.Google Scholar
[19] Birkenmeier, G. F., Müller, B. J., and Rizvi, S. T., Triangular matrix representations of ring extensions. J. Algebra 265(2003), no. 2, 457–477.Google Scholar
[20] Birkenmeier, G. F., Park, J. K., and Rizvi, S. T., Ring hulls and applications. J. Algebra 304(2006), no. 2, 633–665.Google Scholar
[21] Birkenmeier, G. F., Park, J. K., and Rizvi, S. T., Hulls of semiprime rings with applications to C*-algebras. To appear in J. Algebra.Google Scholar
[22] Brown, K. A., The singular ideals of group rings. Quart. J. Math. Oxford 28(1977), no. 109, 41–60.Google Scholar
[23] Camillo, V. P., Costa-Cano, F. J., and Simon, J. J., Relating properties of a ring and its ring of row and column finite matrices. J. Algebra 244(2001), no. 2, 435–449.Google Scholar
[24] Chatters, A. W. and Hajarnavis, C. R., Rings in which every complement right ideal is a direct summand. Quart. J. Math. Oxford 28(1977), no. 109, 61–80.Google Scholar
[25] Clark, W. E., Twisted matrix units semigroup algebras. Duke Math. J. 34(1967), 417–423.Google Scholar
[26] Cohen, M., A Morita context related to finite automorphism groups of rings. Pacific J. Math. 98(1982), no. 1, 37–54.Google Scholar
[27] Davidson, K. R., C*-algebras by example. Fields Inst. Monograph 6, American Mathematical Society, Providence, RI, 1996.Google Scholar
[28] Elliott, G. A., Automorphisms determined by multipliers on ideals of a C*-algebra. J. Functional Analysis 23(1976), no. 1, 1–10.Google Scholar
[29] Fisher, J. W. and Montgomery, S., Semiprime skew group rings. J. Algebra 52(1978), no. 1, 241–247.Google Scholar
[30] Groenewald, N. J., A note on extensions of Baer and P.P.-rings. Publ. Inst. Math. 34(1983), 71–72.Google Scholar
[31] Han, J., Hirano, Y., and Kim, H., Semiprime ore extensions. Comm. Algebra 28(2000), no. 8, 3795–3801.Google Scholar
[32] Harada, M., On modules with extending properties. Osaka J. Math. 19(1982), no. 1, 203–215.Google Scholar
[33] Hirano, Y., On ordered monoid rings over a quasi-Baer ring. Comm. Algebra 29(2001), no. 5, 2089–2095.Google Scholar
[34] Kaplansky, I., Rings of operators. W. A. Benjamin, New York, 1968.Google Scholar
[35] Lam, T. Y., Lectures on modules and rings. Graduate Texts inMathematics 189, Springer-Verlag, New York, 1999.Google Scholar
[36] Lawrence, J., A singular primitive ring. Proc. Amer. Math. Soc. 45(1974), 59–62.Google Scholar
[37] Liu, Z., A note on principally quasi-Baer rings. Comm. Algebra 30(2002), no. 8, 3885–3890.Google Scholar
[38] Louden, K., Maximal quotient rings of ring extensions. Pacific J. Math. 62(1976), no. 2, 489–496.Google Scholar
[39] Montgomery, S., Outer automorphisms of semi-prime rings. J. London Math. Soc. 18(1978), no. 2, 209–220.Google Scholar
[40] Müller, B. J., The quotient category of a Morita context. J. Algebra 28(1974), 389–407.Google Scholar
[41] Osterburg, J. and Park, J. K., Morita contexts and quotient rings of fixed rings. Houston J. Math. 10(1984), no. 1, 75–80.Google Scholar
[42] Pedersen, G. K., Approximating derivations on ideals of C*-algebras. Invent. Math. 45(1978), no. 3, 299–305.Google Scholar
[43] Pollingher, A. and Zaks, A., On Baer and quasi-Baer rings. Duke Math. J. 37(1970), 127–138.Google Scholar
[44] Rieffel, M. A., Actions of finite groups on C*-algebras. Math. Scand. 47(1980), no. 1, 157–176.Google Scholar
[45] Rizvi, S. T. and Roman, C. S., Baer and quasi-Baer modules. Comm. Algebra 32(2004), no. 1, 103–123.Google Scholar
[46] Yi, Z. and Zhou, Y., Baer and quasi-Baer properties of group rings. J. Austral. Math. Soc. 83(2007), no. 2, 285–296.Google Scholar