# Thread: Confusion about Direct Limit of R-Modules

1. ## Confusion about Direct Limit of R-Modules

Hello, I just have a quick question about direct limits. Let $\{ A_s | s \in S \}$ be a directed system of $R$-modules, where $S$ is a directed set. I want to show that

$\displaystyle\lim_{\longrightarrow} A_s := \displaystle\coprod_{s\in S} A_s/ \sim$

is an $R$-module, where $a \in A_s \sim b \in A_t$ if there exists $u \ge s,t$ such that $f_{su}(a) = f_{tu}(b) \in A_u$.

Now in my notes we define addition as follows:

The sum of $a \in A_s$ and $b \in A_t$ is given by choosing an element $u \in S$ with $u \ge s,t$ and $a+b := f_{su}(a) + f_{tu}(b)$ in the $R$-module $A_u$.

But my question is, surely this thing is not well defined because there might be some $u' \ge s,t$, where $u' \not= u$! So this addition would be in a different $R$-module. Does this limit module only make sense if in our directed set $S$ we make a specific choice of element $u$ for every $(s,t)$ pair in $S$?

Thanks for any help.

2. ## Re: Confusion about Direct Limit of R-Modules

Just after posting this I think I've worked it out. If we supposed we have $u'$ such that $u' \ge s,t$, then choose $T \ge u,u' \ge s,t$. Then $f_{uT}(f_{su}(a)) = f_{sT}(a) = f_{u' T}(f_{su'}(a))$. Therefore $f_{su}(a) \in A_u \sim f_{su'}(a) \in A_{u'}$. By a similar argument we have $f_{tu}(b) \in A_u \sim f_{tu'}(b) \in A_{u'}$.

Therefore in the direct limit, $a+b$ is the same no matter which $u$ we pick. $\Box$