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A notion of 2-primal rings is generalized to modules by defining 2-primal modules. We show that the implications between rings which are reduced, have insertion-of-factor-property (IFP), reversible, semi-symmetric and 2-primal are preserved when the notions are extended to modules. Like for rings, 2-primal modules bridge the gap between modules over commutative rings and modules over non-commutative rings; for instance, for 2-primal modules, prime submodules coincide with completely prime submodules. Completely prime submodules and reduced modules are both characterized. A generalization of 2-primal modules is done where 2-primal and NI modules are a special case.

We define and characterize classical completely prime submodules which are a generalization of both completely prime ideals in rings and reduced modules (as defined by Lee and Zhou in [18]). A comparison of these submodules with other "prime" submodules in literature is done. If Rad(M ) is the Jacobson radical of M and β c cl (M ) the classical completely prime radical of M , we show that for modules over left Artinian rings R, Rad(M ) ⊆ β c cl (M ) and Rad( R R) = β c cl ( R R).