Semiprime and weakly compressible modules

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 0 not equal N <= M-R). 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-R is weakly compressible (resp. semiprime) if and only if M is an element of Cog (Soc(M) circle plus R) and M / Soc (M) is an element of Cog (R) (resp. M is an element of Cog (Soc(M) circle plus R)). These considerations settle two questions, namely Qu 1, and Qu 2, in [6, p 92].