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Abstract
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An R-module M is called semiprime (resp. weakly compressible) if it is cogenerated by each of its essential submodules (resp. Hom_{R}(M,N)N is nonzero for every nonzero submodule N of M). We carry out a study of weakly
compressible (semiprime) modules and show that there exist semiprime
modules which are not weakly compressible. Weakly compressible modules with enough critical submodules are characterized in different ways.
For certain rings R, including prime hereditary Noetherian rings, it
is proved that M is weakly compressible (resp. semiprime) if and
only if M \in Cog(Soc(M)\oplus R) and M/Soc(M)\in Cog(R) (resp. M\in Cog(Soc(M) \oplus R)). These considerations settle two open problems , namely Q1 and Q2 in [6. p. 92]
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