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 an R-homomorphism f' : R -> E. In this paper, we first construct an example to show that maximal injectivity is a proper generalization of injectivity. Then we prove that any right R-module over a left perfect ring R is maximally injective if and only if it is injective. We also give a partial affirmative answer to Faith's conjecture by further investigating the property of maximally injective rings. Finally, we get an approximation to Faith's conjecture, which asserts that every injective right R-module over any left perfect right self-injective ring R is the injective hull of a projective submodule.