Hello, I just have a quick question about direct limits. Let be a directed system of -modules, where is a directed set. I want to show that

is an -module, where if there exists such that .

Now in my notes we define addition as follows:

The sum of and is given by choosing an element with and in the -module .

But my question is, surely this thing is not well defined because there might be some , where ! So this addition would be in a different -module. Does this limit module only make sense if in our directed set we make a specific choice of element for every pair in ?

Thanks for any help.