Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T10:37:03.669Z Has data issue: false hasContentIssue false
  • English
  • Français

On uniformly strongly prime gamma rings

Published online by Cambridge University Press:  17 April 2009

G.L. Booth  and
N.J. Groenwald
G.L. Booth
Affiliation:
Department of Mathematics, University of Zululand, Private Bag X1001, 3886 Kwadlangezwa, Republic of South Africa
N.J. Groenwald
Affiliation:
Department of Mathematics, University of Port Elizabeth, P.O. Box 1600, 6000 Port Elizabeth, Republic of South Africa
Rights & Permissions [Opens in a new window]

Abstract

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.

The concept of uniformly strongly prime (usp) is introduced for Γ-ring, and a usp radical τ(M) is defined for a Γ-ring M. If M has left and right unities, then τ(L)+ = τ(M) = τ(R)*, where L and R denote, respectively, the left and right operator rings of M, and τ(·) denotes the usp radical of a ring. If m, n are positive integers, then τ(Mmn) = (τ(M))mn, where Mmn is the matrix Γnm-ring. τ is shown to be a special radical in the variety of Γ-rings. τ1 is the upper radical determined by the class of usp Γ-rings of bound 1. τ ⊆ τ1, but the reverse inclusion does not hold in general. The place of τ and τ1 in the hierarchy of radicals for Γ-rings is shown.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 3 , June 1988 , pp. 437 - 445
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Booth, G.L., ‘A Brown-McCoy radical for Γ-rings’, Quaestiones Math. 7 (1984), 251–262.CrossRefGoogle Scholar
[2]Booth, G.L., ‘Operator rings of a gamma ring’, Math. Japon. 31 (1986), 175–183.Google Scholar
[3]Booth, G.L., ‘Supernilpotent radicals of Γ-rings’, Acta Math. Hangar. (to appear).Google Scholar
[4]Booth, G.L., ‘The strongly prime radicals of a gamma ring’, (submitted).Google Scholar
[5]Booth, G.L. and Groenewald, N.J., ‘On strongly prime and superprime gamma rings’, Ann. Univ. Sci. Budapest (to appear).Google Scholar
[6]Coppage, W.E. and Luh, J.., ‘Radicals of gamma rings’, J. Math. Soc. Japan 23 (1971), 40–52.CrossRefGoogle Scholar
[7]Heyman, G.A.P. and Roos, C., ‘Essential extensions in Radical theory for rings’, J. Austral. Math. Soc. Ser A 23 (1977), 340–347.CrossRefGoogle Scholar
[8]Kyuno, S., ‘On prime gamma rings’, Pacific J. Math. 75 (1978), 185–190.CrossRefGoogle Scholar
[9]Kyuno, S., ‘A gamma ring with the left and right unities,’, Math. Japon 24 (1979), 191–193.Google Scholar
[10]Kyuno, S., ‘Prime ideals in gamma rings’, Pacific J. Math. 98 (1982), 375–379.CrossRefGoogle Scholar
[11]Olson, D.M., ‘A uniformly strongly prime radical’, J. Austral. Math. Soc. Ser. A (to appear).Google Scholar
[12]Olson, D.M. and Veldsman, S., ‘Some remarks on uniformly strongly prime radicals’, (submitted).Google Scholar
[13]Raftery, J.G., M.Sc. dissertation, in (University of Natal, 1985).Google Scholar
[14]Sands, A.D., ‘Radicals and Morita contexts’, J. Algebra 24 (1973), 335–345.CrossRefGoogle Scholar