  Different and distinct notions of regularity for modules exist in the
literature. When these notions are restricted to commutative rings, they all
coincide with the well-known von-Neumann regularity for rings. We give new
characterizations of these distinct notions for modules in terms of both
(weakly-)morphic modules and reduced modules. Furthermore, module theoretic
settings are established where these in general distinct notions turn out to be
indistinguishable.
