On the flatness and injectivity of dual modules (Q1281305)

If \(E\) is an injective cogenerator in the category of \(R\)-modules, for a commutative ring \(R\), \(M^e=\Hom_R(M,E)\) denotes the dual module of an \(R\)-module \(M\). The author in this paper derives some necessary and sufficient conditions for the \(FP\)-injectivity and injectivity of an \(R\)-module \(M\) in terms of its dual module \(M^e\) for coherent rings, Noetherian rings and Artinian rings. Though the paper is quite interesting, there are two minor slips. In the proof of \((1)\to(3)\) of lemma 1, on p. 258, in the top row, the extreme right arrow `\(\to 0\)' is to be deleted as the author proves that. In the proof of \((5)\to(1)\) of theorem 1 the author uses the fact that purity is closed under taking arbitrary direct products which may not be true as tensor product need not commute with arbitrary direct products.