In this paper we study prime preradicals, irreducible preradicals, ∧-prime preradicals, prime submodules and diuniform modules. We study some relations between these concepts, using the lattice structure of preradicals developed in previous papers. In particular, we give a characterization of prime preradicals using an operator named the relative annihilator. We also characterize prime submodules by means of prime preradicals. We give some characterizations of rings that have certain conditions on prime radicals and on irreducible preradicals, such as left local left V-rings, as well as 1-spr rings, which we introduce.

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We continue the study of the preradicals of a ring in the lattice point of view. We introduce several interesting preradicals associated to a given preradical and some partitions of the whole lattice in terms of preradicals. As an application, we also give some classification theorems.

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