Copolyform modules (Q2731565)

Let \(R\) be a ring with identity and \(M\) a unital right \(R\)-module. \(M\) is called copolyform if for any small submodule \(N\) of \(M\), \(\Hom_R(M,N/K)=0\) for all submodules \(K\) of \(N\). Small submodules of rationals are studied first. It is shown that \(\mathbb{Q}\) is copolyform regarded as a \(\mathbb{Z}\)-module.NEWLINENEWLINENEWLINESome samples of further results: (1) Direct summands of copolyform modules are copolyform. (2) If \(x\in\text{Rad}(M)\), where \(M\) is an \(R\)-module and \(M/xR\) has a projective cover, then \(M\) is copolyform if and only if \(M/xR\) is copolyform. (3) If \(M\) is a projective \(R\)-module with endomorphism ring \(\text{End}(M)=S\), then \(M\) is copolyform if and only if \(S\) is copolyform. (4) Modules over \(V\)-rings are copolyform.