Modules with many homomorphisms

It is proved that if R is a right FBN ring then a non-zero right R-module M has the property that HomR(M,N)≠0 for every non-zero submodule N of M if and only if HomR(M,R/P)≠0 for every associated prime ideal P of M. One consequence is that over a commutative Noetherian ring R, HomR(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 HomR(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 HomR(M,R)≠0 then HomR(M,M′)≠0 for every non-zero R-module M′. It is shown that this result does not extend to prime Noetherian PI rings.