It seems to me the references in this [Mathematics - Stack Exchange answer](http://math.stackexchange.com/questions/82957/preservation-of-direct-sums-and-finite-generation/82958#82958) contain the requested information. 

**EDIT.** Here is an excerpt from Hyman Bass's book **Algebraic K-Theory**, W. A. Benjamin (1968), p. 54: 

> Exercise. 

> (a) Show that a module $P$ is finitely generated if and only if the union of a totally ordered family of proper submodules of $P$ is a proper submodule.

> (b) Show that $\text{Hom}_A(P,\bullet)$ preserves coproducts if and only if the union of every (countable) chain of proper submodules is a proper submodule.

> (c) Show that the conditions in (a) and (b) are not equivalent. (Examples are not easy to find.)