J Austral Math Soc Ser A 53 pp55--63, 1992.

On Priime One-Sided Ideals, Bi-Ideals and Quasi-Ideals of a Gamma Ring

G. L. Booth and N. J. Groenewald

Abstract

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

1991 AMS Subject Classification: 16A78

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Authors

G. L. Booth

University of Transkei, Private Bag XI, Umtata, Transkei, South Africa.

N. J. Groenewald

University of Pero Elizabeth, P.O. Box 1600, Port Elizabeth, 6000, South Africa.