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Structure of Γ-rings

1979; Mathematical Sciences Publishers; Volume: 80; Issue: 2 Linguagem: Inglês

10.2140/pjm.1979.80.537

1945-5844

T. S. Ravisankar, U. Shukla,

Fuzzy and Soft Set Theory

In the first part of the present paper, Γ-rings are studied in the setting of modules.The notion of a module over a Turing 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 jR-ring Γ also in the case of a strongly semisimple strongly right Artinian weak Γ N ring.The Jacobson radical of a weak i~Vring 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 /Vrings.The present approach provides a uniform module cum radical theory not only for F-rings, but also for the associative triple systems.The second part of the paper imbeds any weak Γ N -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 Tarings, characterizes the strongly simple, strongly right Artinian weak /Vrings as the Γ-rings of rectangular matrices over division rings.The ring of all square matrices over a division ring plays a vital role in classical ring theory.However, when one considers the set of all rectangular matrices (of the same type), there appears to be no natural way of introducing a binary ring multiplication into it.Various authors like Nobusawa [15], Lister [8] and Hestenes (see [5]) have tried to offset this difficulty by considering a natural ternary multiplication in the set of rectangular matrices; their investigations have led to the respective notions of a Γ^-ring, associative triple systems of first kind (ternary rings) and of second kind.These three structures provide a suitable setting for the study of rectangular matrices.The above-mentioned authors have obtained some structural results for these structures, results similar to ones for rings.The concept of weak Γ N -τing introduced in this paper includes all the three above structures, besides rings, as particular cases.Nobusawa considers a notion of semisimplicity for his Γ^-ring and that does not arise from a radical as in the case of rings.Coppage and Luh [2] have considered a few radicals among which