Modules with many homomorphisms

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It is proved that if R is a right FBN ring then a non-zero right R-module M has the property that Hom R ( M , N ) ≠ 0 for every non-zero submodule N of M if and only if Hom R ( M , R / P ) ≠ 0 for every associated prime ideal P of M. One consequence is that over a commutative Noetherian ring R, Hom R ( X , Y ) ≠ 0 for every non-zero projective R-module X and every non-zero submodule Y of X. In case R is a left Noetherian right FBN ring, then a non-zero finitely generated right R-module M has the property that Hom R ( M , N ) ≠ 0 for every non-zero submodule N of M if and only if the right ( R / P ) -module M / MP is not torsion for every associated prime ideal P of M. Finally, if R is a commutative Noetherian ring and M is an R-module such that Hom R ( M , R ) ≠ 0 then Hom R ( M , M ′ ) ≠ 0 for every non-zero R-module M ′ . It is shown that this result does not extend to prime Noetherian PI rings.