The rings discussed in this paper will have an identity element and will be associative. Similarly, unless specified otherwise, all modules will be right modules. A submodule L of V is considered essential in V if it does not contain a nonzero element from an arbitrary nonzero submodule U of V . On the other hand, a submodule W of V is considered small in V if V is not equal to the sum of any proper submodule L of V with W. The module V is hollow, provided each proper submodule is small in V . For a module V and W ≤ V , the notion W ≤⊕ V means W is a direct summand (ds, for short) of V . Note also that Rad(V ) is defined to be the sum of all small submodules of V and Soc(V ) is equal to sum of all simple submodules of V .
