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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

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