Some Results on Simple-Direct-Injective Modules

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.