Let \(A\) and \(B\) be modules over a commutative ring \(R\), and let \(\pi(A,B)\) denote the image of the evaluation map of \(\Hom (A,B) \otimes A\) into \(B\). Let \(T(A)\) denote \(\pi(A,R)\), the trace ideal of \(A\). The authors observe that \(\pi(A,B) =T(A) \cdot B\) whenever \(T(A) \cdot A=A\), in which case \(\pi(A,B)\) is a pure submodule of \(B\) if \(T(A)\) is a pure submodule of \(R\). The authors claim that the additional hypothesis of \(T(A)\) being a pure ideal is fulfilled when \(A\) is a projective module or \(A\) is a multiplication module, i.e. every submodule of \(A\) has the form \(I\cdot A\) for some ideal \(I\) of \(A\). A proof of the claim for multiplication modules will appear in a paper submitted for publication by the first author. Next let \(F(H)\) denote the image of the canonical homomorphism of \(\Hom (A,R) \otimes B\) into \(H= \Hom (A,B)\). If \(B\) is a projective module or a multiplication module with \(T(B) \cdot B=B\), the authors prove that the elements of \(F(H)\) are the homomorphisms \(f\) for which \(f(A)\) is contained in a finitely generated submodule of \(B\), \(F(H)\) is dense in \(H\), and \(F(H)\) is a pure submodule of \(H\). The proof for multiplication modules, theorem 2.8 of this paper, is marred by misprints and misstatements.