The existence of envelopes (Q1315167)

For a ring \(R\) and a class \(\mathcal F\) of left \(R\)-modules an \(\mathcal F\)- preenvelope of a left \(R\)-module \(M\) is a linear map \(\phi : M \to F\), \(F \in {\mathcal F}\), such that for any linear map \(f: M \to F'\), \(F' \in {\mathcal F}\), there is a \(g : F \to F'\) such that \(g \circ \phi = f\). If when \(F = F'\) and \(f = \phi\) the only such completing maps are automorphisms of \(F\), then \(F\) is said to be an \(\mathcal F\)-envelope of \(M\). \(\mathcal F\)- envelopes are unique up to isomorphism when they exist. \textit{J.-M. Maranda} [Trans. Am. Math. Soc. 110, 98-135 (1964; Zbl 0121.266)] defined an injective structure \(({\mathcal A,\mathcal F})\) in the category of left \(R\)-modules to be a class \(\mathcal A\) of linear mappings between left \(R\)-modules and a class \(\mathcal F\) of left \(R\)-modules such that: 1) \(F \in {\mathcal F}\) if and only if \(\text{Hom}_ R(N,F) \to \text{Hom}_ R(M,F) \to 0\) is exact for all \(M \to N \in {\mathcal A}\), 2) \(M \to N \in {\mathcal A}\) if and only if \(\text{Hom}_ R(N,F) \to \text{Hom}_ R(M,F) \to 0\) is exact for all \(F\in {\mathcal F}\), and 3) Every left \(R\)-module \(M\) has an \(\mathcal F\)-preenvelope \(M \to F\). A class \(\mathcal G\) of right \(R\)-modules is said to determine the injective structure \((\mathcal A,\mathcal F)\) if \(M \to N \in {\mathcal A}\) if and only if \(0 \to G \otimes M \to G \otimes N\) is exact for all \(G \in {\mathcal G}\). The authors show that every left \(R\)-module has an \(\mathcal F\)-envelope relative to an injective structure \(({\mathcal A,\mathcal F})\) determined by a class \(\mathcal G\) of right \(R\)-modules. Moreover, they show that if \(\mathcal G\) is any set of right \(R\)-modules, there is a unique injective structure \(({\mathcal A},{\mathcal F})\) determined by \(\mathcal G\). In fact, they show that \(F \in {\mathcal F}\) if and only if \(F\) is isomorphic to a direct summand of products of copies of the character modules \(G^ + = \text{Hom}_ Z (G,Q/Z)\), \(G \in {\mathcal G}\). Using these results the authors go on to show that if \(R\) is right coherent, then every left \(R\)-module has a pure injective envelope and when \(R\) is right coherent and pure injective as a left \(R\)-module, they show that every finitely presented left \(R\)-module has a flat envelope.