D-Extending Modules

A submodule $N$ of a module $M$ is called d-closed if $M/N$ has a zero socle. D-closed submodules are similar concept to s-closed submodules, which are defined through nonsingular modules by Goodearl. In this article we deal with modules with the property that all d-closed submodules are direct summands insert ignore into journalissuearticles values(respectively, closed, pure);. The structure of a ring over which d-closed submodules of every module are direct summand insert ignore into journalissuearticles values(respectively, closed, pure); is studied.