Injectivity Relative to Closed Submodules

Journal of Algebra, 321(2) (2009), 548-557
http://dx.doi.org/10.1016/j.jalgebra.2008.11.004

Let R be a ring. An R-module X is called c-injective if, for every closed submodule L of every R-module M, every homomorphism from L to X lifts to M. It is proved that if R is a Dedekind domain then an R-module X is c-injective if and only if X is isomorphic to a direct product of homogeneous semisimple R-modules and injective R-modules. It is also proved that a commutative Noetherian domain R is Dedekind if and only if every simple R-module is c-injective.