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.
Thank you in advance. Cheers
Rotman's book: "An Introduction to Homological Algebra" has:
And