# Direct limit of modules: a property.

Suppose $A$ to be a ring and $M_i$ the indexed $A$-modules used to build the direct limit of modules $M \doteq \lim{M_i}$. Let $f_{ij}: M_i \to M_j$ the transition maps and $\phi : M_i \to M$ the projection map.

Show that every element in $M$ is of the form $\phi(m_i)$ for an $m_i \in M_i \subseteq \oplus_{i \in I}{M_i}$

My attempt follows. Let $m \in M$, then is can be written as $[(m_1,m_2,....m_n,0,0,0,..)]$ (finite non-zero components - I wrote it in that way just for simplicity). In order to be written as a $\phi(m_k)$, I want to see that its class is equal to $[(0,0,0,..,m_k,0,...)]$, in other words there exists a $t$ such that $f_{1t}(m_1)=....=f_{hn}(m_n)$ - but I simply don't see: why is it true? I am quite sure that I am misunderstanding something.

• You seem to use some very ugly "definition" of the colimit? Commented May 7, 2015 at 9:50
• Dear @MartinBrandenburg, I am not sure to have understood the meaning of your comment (sorry - my English... :) ). In any case, I use the definition given by wikipedia.
– user233650
Commented May 7, 2015 at 9:52
• can you use Def. of Rotman? With Def. of Rotman, this is "Lemma 5.30" of Rotman. Commented May 7, 2015 at 10:01
• @user1 I don't have the Rotman's book :( Exact title?
– user233650
Commented May 7, 2015 at 10:05
• i dont know how to post a pic. in comment. would you like it as an answer? Commented May 7, 2015 at 10:07