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.
