  The coincidence of the set of all nilpotent elements of a ring with its prime
radical has a module analogue which occurs when the zero submodule satisfies
the radical formula. A ring $R$ is 2-primal if the set of all nilpotent
elements of $R$ coincides with its prime radical. This fact motivates our study
in this paper, namely, to compare 2-primal submodules and submodules that
satisfy the radical formula. A demonstration of the importance of 2-primal
modules in bridging the gap between modules over commutative rings and modules
over noncommutative rings is done and new examples of rings and modules that
satisfy the radical formula are also given.
