Partial endomorphisms of almost self injective modules

Let M be a quasi-injective module and S be its ring of endomorphisms. It is well known that the radical J of S is the set of all those endomorphisms of M which have large kernels, and S/J is a von Neumann regular ring. Such a result does not hold for almost self-injective modules. Let N be an almost self-injective module and G its ring of endomorphisms. By enlarging the ring of endomorphisms of N, it is shown that a generalization of the above result for quasi-injective modules can be given for N. In addition an artinian serial ring R with no non-trivial central idempotent is studied. A characterization for R to be almost right self-injective in terms of its Kuppisch series is given, which can help in creating certain types of almost self-injective rings.