We give the new results on the theory of the one-sided (left) strongly prime modules and their strongly prime radicals. Particularly, the conceptually new and short proof of the A.L.Rosenberg’s theorem about one-sided strongly prime radical of the ring is given. Main results of the paper are: presentation of each left stongly prime ideal p of a ring R as p = R ∩ M, where M is a maximal left ideal in a ring of polynomials over the ring R; characterization of the primeless modules and characterization of the left strongly prime radical of a finitely generated module M in terms of the Jacobson radicals of modules of polynomes M(X1, . . . , Xni) .
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