This is one of a series of papers on complete rings of linear transformations by the author. It depends heavily on some previous results by the author. Suppose that $M$ is a finite-dimensional vector space over a division ring $F$. Then its conjugate space $M^*$ is also finite-dimensional. There is a simple correspondence of linear transformations in one to those of the other via transpose of matrices. For infinite vector spaces, this correspondence breaks down. The author proposes the following remedy: Let $M$ be a vector space of arbitrary dimension over a division ring $F$. Let $\Omega$ be a complete ring of linear transformations on $M$. Let $A=E\Omega$ be a minimum right ideal of $\Omega$, with $E^2=E$. Let $K=E\Omega E$. It is known that $K$ is a division ring. From earlier results of the author, considering $A$ as a vector space over $K$, $\Omega$ is a complete ring of $K$-linear transformations of $A$. Now, let $A^*=\Omega E$. Again, by previous results of the author, $A$ and $A^*$ are ``conjugate'' to each other; further, there is a correspondence between $A^*$ as a two-sided $(\Omega,F)$-module and $M^*$ as a two-sided $(\Omega,K)$-module. By use of this correspondence, the author proves an embedding theorem of $\Omega$ in the second section of the paper. In the third and final section a structure theorem of primitive rings is presented. A rather long sequence of definitions and notation as well as dependence of earlier results prevent a more detailed summary of these interesting results. Hopefully, a more self-contained treatment in a more accessible language will help to make them better known.