Some Notes on Maximal Injectivity

A right R-module E over a ring R is said to be maximally injective in case for any maximal right ideal m of R,every R-homomorphism f:m→E can be extended to a R-homomorphism f′∶R→E.In this paper,the authors first extend the conception of injective dimensions of any ring R to that of maximally injective dimensions,which turns out to be precisely the supremum of projective dimensions of simple R-modules.Then they investigate modules with maximally injective character modules,which has inherent connection with an famous open problem of Ramamurthi concerning Von Neumann regular rings.They end this paper with a partial affirmative answer to Ramamurthi's problem under the right socular assumption.