I asked this [question](http://math.stackexchange.com/questions/82957/preservation-of-direct-sums-and-finite-generation/) on Mathematics - Stack Exchange (MSE). Having figured out out how to handle the problem in an **extremely** particular case, I also posted it as an [answer](http://math.stackexchange.com/questions/82957/preservation-of-direct-sums-and-finite-generation/82958#82958) (in the technical sense of the term) to my own question. Getting no other answer, I thought I could post the question on MathOverflow. For the reader's convenience, here is a copy and paste of the question. 

This is a follow up on this MSE [question](http://math.stackexchange.com/questions/78161/hom-and-direct-sums), asked by Evariste. 

Let $R$ be an associative ring with one. The word "module" shall mean *left* $R$-module. Say that a module $A$ **preserves direct sums** if the functor $\hom_R(A,?)$ does.  

The **main question** is

> Does the condition that $A$ preserves direct sums imply that $A$ is finitely generated? 

The converse is clear: see this MSE [answer](http://math.stackexchange.com/questions/78161/hom-and-direct-sums/78178#78178). 

As observed by Mariano Suárez-Alvarez in a comment to this MSE [answer](http://math.stackexchange.com/questions/78161/hom-and-direct-sums/78178#78178), if $A$ can be written as the union of an increasing sequence $(A_n)_{n\in\mathbb N}$ of submodules, then $A$ does **not** preserve direct sums. [The argument is described in the answer.]

Say that $A$ is **countably cofinal** if it can be written as such a union. If $A$ is neither finitely generated nor countably cofinal, say that $A$ is **uncountably cofinal**. 

[Here is the motivation for this terminology. A group which can be written as the union of an increasing sequence of subgroups is called *countably cofinal*, and a group which is neither finitely generated nor countably cofinal, is called *uncountably cofinal*. Uncountably cofinal groups have been studied by Serre, Tits, MacPherson, Bergman, and many others: see this [Google Search](http://www.google.com/search?q=%22uncountable+cofinality%22&hl=en&safe=off#sclient=psy-ab&hl=en&safe=off&source=hp&q=%22uncountable+cofinality%22+serre+tits+macpherson+bergman&pbx=1&oq=%22uncountable+cofinality%22+serre+tits+macpherson+bergman&aq=f&aqi=&aql=&gs_sm=e&gs_upl=51552l58613l2l59788l4l3l1l0l0l0l492l1142l2-1.0.2l4l0&bav=on.2,or.r_gc.r_pw.r_cp.,cf.osb&fp=5a3ef425d86707e8&biw=1422&bih=705). In particular, uncountably cofinal groups do exist.] 

The **second question** is: 

> Do uncountably cofinal modules exist?