| Abstract: | In this paper, we generalize the classical structure theorem and isomorphism
theorem of primitive rings. Let u be a primitive ring, then u can be regarded as a
dense subring of linear transformations in a vector space M. Lei \Omega be the ring of all linear transformations of M, we define (BV=^TV\ 11, where Tv is the ideal of Q which contains all linear transformations t, with rank K X V, If is ^-transitive,we call the z^-socle of 11.
If U is a primitve ring with r-socle , then we can find a pair of ^-modules
(JT, whereJf3 is a subring of and we can define a dual topology in
such that 11 is contained in the ring of all continuous endormorphisms
of . We obtained an inclusion relation ?=S?(3ft, 2ft') 3J?7 j^')
which refines the relation due to Jacobson: J§f(3ft, 2ft') iDllQ^o'133. We also proved that srf and are uniquely determined by 11 within semi-module isomorphism. In the last section, we obtained a structure theorem of finitely generated one-sided ideals of?,. |
