A module M is called a simple-direct-injective module if, whenever A and B are simple submodules of M with A≅B and B is a direct summand of M, then A is a direct summand of M. Some new characterizations of these modules are proved. The structure of simple-direct-injective modules over a commutative Dedekind domain is fully determined. Also, some relevant counterexamples are indicated to show that a left simple-direct-injective ring need not be right simple-direct-injective.

Keywords: Dedekind domain, Simple-Direct-Injective module, Left (Right) simple-direct-injective ring.