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On prime one-sided ideals, bi-ideals and quasi-ideals of a gamma ring

Published online by Cambridge University Press:  09 April 2009

G. L. Booth  and
N. J. Groenewald
G. L. Booth
Affiliation:
University of TranskeiPrivate Bag XI Umtata Transkei, South Africa
N. J. Groenewald
Affiliation:
University of Port ElizabethP.O. Box 1600 Port Elizabeth 6000, South Africa
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Abstract

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Let M be a Γ-ring with right operator ring R. We define one-sided ideals of M and show that there is a one-to-one correspondence between the prime left ideals of M and R and hence that the prime radical of M is the intersection of its prime left ideals. It is shown that if M has left and right unities, then M is left Noetherian if and only if every prime left ideal of M is finitely generated, thus extending a result of Michler for rings to Γ-rings.

Bi-ideals and quasi-ideals of M are defined, and their relationships with corresponding structures in R are established. Analogies of various results for rings are obtained for Γ-rings. In particular we show that M is regular if and only if every bi-ideal of M is semi-prime.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 1 , August 1992 , pp. 55 - 63
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Andrunakievich, A. V. and Andrunakievich, V. A., ‘One sided ideals and radicals of rings’, Algebra i Logika 20 (1981), 489–510.Google Scholar
[2]Chen, W. X., ‘The largest von Neumann regular ideal of a γ-ring’, Xhejiang Dascue Xueboa 18 (1984), 4, 133–138 (in Chinese).Google Scholar
[3]Coppage, W. E. and Luh, J., ‘Radicals of gamma rings’, J. Math. Soc. Japan 23 (1971), 40–52.CrossRefGoogle Scholar
[4]Kyuno, S., ‘A gamma ring with the right and left unities’, Math. Japan. 24 (1979), 191–193.Google Scholar
[5]Yuno, S., ‘Prime ideals in gamma rings’, Pacific J. Math. 98 (1982), 375–379.Google Scholar
[6]Michler, G. O., ‘Prime right ideals and right noetherian rings’, in Ring Theory, edited by Gordon, R., Academic Press, London (1972).Google Scholar
[7]Steinfield, O., ‘Quasi-ideals in rings and semigroups’, Akademiai Kiado Budapest (1978).Google Scholar
[8]van der Walt, A. P. J., ‘Prime and semiprime bi-ideals’, Quaestiones Mathematicae 5 (1983).CrossRefGoogle Scholar