Vol. 80, No. 2, 1979

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Structure of Γ-rings

T. S. Ravisankar and U. S. Shukla

Vol. 80 (1979), No. 2, 537–559

In the first part of the present paper, Γ-rings are studied in the setting of modules. The notion of a module over a Γ-ring is studied, with the object of developing the notion of a Jacobson-radical for a Γ-ring via modules. This radical enjoys the usual properties of the corresponding object in rings. A semisimple right Artinian Γ-ring turns out to be the direct sum of simple ideals; this conclusion is strengthened to include a corresponding decomposition for the R-ring Γ also in the case of a strongly semisimple strongly right Artinian weak ΓN-ring. The Jacobson radical of a weak ΓN-ring R is characterized in different ways, in one of them as the set of all properly quasi-invertible elements of R. It is shown how rings, ternary rings and associative triple systems can be considered as weak ΓN-rings. The present approach provides a uniform module cum radical theory not only for Γ-rings, but also for the associative triple systems.

The second part of the paper imbeds any weak ΓN-ring R into a suitable associative ring A. Simplicity and semisimplicity in R and A are shown to be related. The main result of this part which generalizes the classical Wedderburn-Artin theorem for rings to Γ-rings, characterizes the strongly simple, strongly right Artinian weak ΓN-rings as the Γ-rings of rectangular matrices over division rings.

Mathematical Subject Classification
Primary: 16A78, 16A78
Received: 26 November 1975
Published: 1 February 1979
T. S. Ravisankar
U. S. Shukla