Throughout \(R\) is an associative ring with identity and all modules are unitary right \(R\)-modules. For a given module \(L\), a module \(M\) is called \(L\)-simple-injective (resp. \(L\)-FI-injective) if for any submodule \(K\) of \(L\), any homomorphism \(K\to M\) with image simple (resp. finitely generated) can be extended to a homomorphism \(L\to M\). A module \(M\) is called semicompact if any finitely solvable system \((x_i,X_i)_{i\in I}\) of \(M\) with \(X_i=l_M(A_i)\) for some \(A_i\subseteq R\) is solvable, where \(l_M(A_i)=\{x\in M\mid xA_i=0\}\).
For an \(R\)-simple-injective module \(M\) with essential socle, the authors establish some conditions for \(M\) to be \(R\)-FI-injective and show that \(M\) is injective if and only if \(M\) is semicompact.