On the flatness and injectivity of dual modules. II (Q2781726)

[For part I see \textit{Z. Huang}, Southeast Asian Bull. Math. 21, No. 3, 257-262 (1997; Zbl 0922.13006).]NEWLINENEWLINENEWLINEFor a commutative ring \(R\) and an injective cogenerator \(E\) in the category of \(R\)-modules, the authors characterize QF rings, IF rings and semihereditary rings in terms of dual modules with respect to \(E\). Actually they prove the following: (i) \(R\) is a QF (resp. IF) ring if and only if for any free (or projective or flat) module \(M\), its dual module with respect to \(E\) is a submodule of a projective (resp. flat) module; (ii) \(R\) is a semihereditary ring if and only if the dual module (with respect to \(E\)) of any finitely presented module is FP-injective. They also define \(E\)-Artin and \(E\)-coherent rings and prove that \(R\) is a QF (resp. IF) ring if and only if \(R\) is an \(E\)-Artin (resp. \(E\)-coherent) and self FP-injective ring.