91Ó°ÊÓ

Let \(E\) be a left \(R\)-module. Prove that \(E\) is injective if and only if \(\operatorname{Ext}_{R}^{1}(A, E)=\\{0\\}\) for every left \(R\)-module \(A\).

(i) Prove that \(e(C, \square): R\) Mod \(\rightarrow\) Ab is a covariant functor if, for \(h: A \rightarrow A^{\prime}\), we define \(h_{*}: e(C, A) \rightarrow e\left(C, A^{\prime}\right)\) by \([\xi] \mapsto[h \xi] .\) (ii) Prove that \(e(C, \square)\) is naturally isomorphic to Ext \(_{R}^{1}(C, \square)\).