Hostname: page-component-6bb9c88b65-9c7xm Total loading time: 0 Render date: 2025-07-23T00:15:44.584Z Has data issue: false hasContentIssue false
  • English
  • Français

Prime essential rings

Published online by Cambridge University Press:  20 January 2009

B. J. Gardner  and
P. N. Stewart
B. J. Gardner
Affiliation:
Mathematics DepartmentUniversity of TasmaniaHobart, Tasmania 7001Australia
P. N. Stewart
Affiliation:
Department of Mathematics, Statistics and Computing ScienceDalhousie UniversityHalifax, Nova ScotiaCanadaB3H 3J5
Rights & Permissions [Opens in a new window]

Extract

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 prime essential if R is semiprime and for each prime ideal P of R, P ∩ I ≠0 whenever I is a nonzero two-sided ideal of R. Examples of prime essential rings include rings of continuous functions and infinite products modulo infinite sums. We show that the class of prime essential rings is closed under many familiar operations; in particular, we consider polynomial rings, matix rings, fixed rings and skew group rings. Also, we explore the relationship between prime essential rings and special radical classes, and we demonstrate how prime essential rings can be used to construct radical classes which are not special.

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 2 , June 1991 , pp. 241 - 250
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Amitsur, S. A., Prime rings having polynomial identities with arbitrary coefficients, Proc. London Math. Soc. 17 (1967), 470–486.Google Scholar
2.Anderson, T., Kaarli, K. and Wiegandt, R., Radicals and subdirect decomposition, Comm. Algebra 13 (1985), 479–494.Google Scholar
3.Beidar, K. I. and Salavova, K., Some examples of supernilpotent nonspecial radicals, Acta Math. Acad. Sci. Hung. 40 (1982), 109–112.Google Scholar
4.Bergman, G. M. and Isaacs, I. M., Rings with fixed point free group actions, Proc. London Math. Soc. 27 (1973), 69–87.Google Scholar
5.Burris, S., Free algebras as subdirect products, Algebra Universalis 11 (1980), 133–134.Google Scholar
6.Chick, H. L. and Gardner, B. J., The preservation of some ring properties by semilattice sums, Comm. Algebra 15 (1987), 1017–1038.Google Scholar
7.Cohen, M. and Montgomery, S., Group graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), 237–258.Google Scholar
8.Divinsky, N., Rings and Radicals (University of Toronto Press, Toronto, 1965).Google Scholar
9.Fisher, J. W. and Montgomery, S. M., Semiprime skew group rings, J. Algebra 52 (1978), 241–247.CrossRefGoogle Scholar
10.Gardner, B. J., Radical Theory (Longman/Wiley, Harlow/New York, 1989).Google Scholar
11.Goodearl, K. R., Prime ideals in regular self-injective rings II, J. Pure Appl. Algebra 3 (1973), 357–373.Google Scholar
12.Goodearl, K. R., Von Neumann Regular Rings (Pitman, London, 1979).Google Scholar
13.Kohls, C. W., Prime ideals in rings of continuous rings, Illinois J. Math. 2 (1958), 505–536.Google Scholar
14.van Leeuwen, L. C. A. and Jenkins, T. L., A supernilpotent nonspecial radical class, Bull. Australian Math. Soc. 9 (1973), 343–348.Google Scholar
15.Lorenz, M.. Finite normalizing extensions of rings, Math. Z. 176 (1981), 447–484.Google Scholar
16.Lorenz, M. and Passman, D. S., Prime ideals in crossed products of finite groups, Israel J. Math. 33 (1979), 89–132.CrossRefGoogle Scholar
17.Matlis, E., The minimal prime spectrum of a reduced ring. Illinois J. Math. 27 (1983), 353–391.CrossRefGoogle Scholar
18.Montgomery, S., Prime ideals in fixed rings, Comm. Algebra 9 (1981), 423–449.Google Scholar
19.Parmenter, M. M. and Stewart, P. N., Excellent extensions, Comm. Algebra 16 (1988), 703–713.Google Scholar
20.Passman, D. S., It's essentially Maschke's theorem, Rocky Mountain J. Math. 13 (1983), 37–54.Google Scholar
21.Ryabukhin, Yu. M., Overnilpotent and special radicals, Studies in Algebra and Math. Anal., Izdat “Karta Moldovenjaske”, Kishinev, 1965, 65–72.Google Scholar
22.Ryabukhin, Yu. M., Supernilpotent and special radicals, Algebra and Modules, Mat. Issled. 48 (1978), 80–93, 169.Google Scholar
23.Rowen, L. H., A subdirect decomposition of semiprime rings and its application to maximal quotient rings, Proc. Amer. Math. Soc. 46 (1974), 176–180.Google Scholar
24.Snider, R. L., Lattices of radicals, Pacific J. Math. 40 (1972), 207–220.Google Scholar
25.Szász, F. A., Radicals of Rings (Budapest, Akadémiai Kiadó, 1981).Google Scholar
26.Wiegandt, R., Radical and Semisimple Classes of Rings (Queen's Papers in Pure and Applied Mathematics, No. 37, Kingston, 1974).Google Scholar