Semiprime and weakly compressible modules

Abstract

An R-module M is called semiprime (resp. weakly compressible) if it is cogenerated by each of its essential submodules (resp. HomR(M, N)N is nonzero for every 0 6= N ≤ MR). 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 MR is weakly compressible (resp. semiprime) if and only if M ∈ Cog(Soc(M) ⊕ R) and M/Soc(M) ∈ Cog(R) (resp. M ∈ Cog(Soc(M) ⊕ R)). These considerations settle two questions, namely Qu 1, and Qu 2, in [6, p 92].