On mono-injective modules and mono-ojective modules. (Q2839629)

Let \(M\) and \(N\) be two modules. Then \(M\) is called generalized (mono-)\(N\)-injective or (mono-)\(N\)-ojective if for any submodule \(X\) of \(N\) and any homomorphism (monomorphism) \(f\colon X\to M\), there exist decompositions \(M=M_1\oplus M_2\) and \(N=N_1\oplus N_2\), a homomorphism (monomorphism) \(g_1\colon N_1\to M_1\) and a monomorphism \(g_2\colon M_2\to N_2\) with the following property: for any \(x\in X\), if one writes \(x=n_1+n_2\in N_1\oplus N_2\) and \(f(x)=m_1+m_2\in M_1\oplus M_2\), then \(g_1(n_1)=m_1\) and \(g_2(m_2)=n_2\). -- The authors study this class of modules, and use it in order to find some necessary and sufficient conditions for direct sums of extending modules to be extending.