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On Primitive Ideals in Graded Rings

Published online by Cambridge University Press:  20 November 2018

Agata Smoktunowicz
Agata Smoktunowicz*
Affiliation:
School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK. e-mail: A.Smoktunowicz@ed.ac.uk
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Abstract

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Let $R\,=\,\oplus _{i=1}^{\infty }\,{{R}_{i}}$ be a graded nil ring. It is shown that primitive ideals in $R$ are homogeneous. Let $A\,=\,\oplus _{i=1}^{\infty }\,{{A}_{i}}$ be a graded non-PI just-infinite dimensional algebra and let $I$ be a prime ideal in $A$. It is shown that either $I\,=\,\{0\}$ or $I\,=\,A$. Moreover, $A$ is either primitive or Jacobson radical.

Keywords

16D6016W50
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 3 , 01 September 2008 , pp. 460 - 466
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Bartholdi, L., Branch rings, thinned rings, tree enveloping rings. Israel J. Math. 154(2006), 93–139.Google Scholar
[2] Divinski, N. J., Rings and Radicals. Mathematical Expositions 14, University of Toronto Press, Toronto, ON, 1965.Google Scholar
[3] Faith, C., Rings and Things and a Fine Array of Twentieth Century Associative Algebra. Mathematical Surveys and Monographs 65, American Mathematical Society, Providence, RI, 2004.Google Scholar
[4] Farkas, D. R., and Small, L. W., Algebras which are nearly finite dimensional and their identities. Israel J. Math. 127(2002), 245–251.Google Scholar
[5] Krause, G. and Lenagan, T. H., Growth of Algebras and Gelfand-Kirillov Dimension. Revised edition. Graduate Studies in Mathematics 22, American Mathematical Society, Providence, RI, 2000.Google Scholar
[6] Lam, T. Y., Exercises in classical ring theory. In: Problem Books in Mathematics, Springer-Verlag, New York, 1995, pp. 122–123.Google Scholar
[7] Reichstein, Z., Rogalski, D., and Zhang, J. J., Projectively simple rings. Adv. Math. 203(2006), no. 2, 365–407.Google Scholar
[8] Small, L. W., Stafford, J. T., and Warfield, R. B. Jr, Affine algebras of Gelfand-Kirillov dimension one are PI. Math. Proc. Cambridge Phil. Soc. 97(1984), no. 3, 407–414.Google Scholar
[9] Smoktunowicz, A., On primitive ideals in polynomial rings over nil rings. Algebr. Represent. Theory 8(2005), no. 1, 69–73.Google Scholar
[10] Smoktunowicz, A. and Puczyłowski, E. R., A polynomial ring that is Jacobson radical but not nil. Israel J. Math. 124(2001), 317–325.Google Scholar
[11] Szász, F. A., Radicals of rings. Translated by the author, Wiley Interscience Chichester, 1981.Google Scholar