Suppose I take a direct limit in the category of groups, or the category of R-modules, or similar. Let $I$ denote the index set, $A_i$ the objects and $f_{ij}: A_i \to A_j$ the structure maps of the direct system. The colimit over this direct system can then be computed via a coequalizer (as e.g. explained in http://en.wikipedia.org/wiki/Direct_limit). Now in general no requirements are made with respect to the structure maps $f_{ij}$, in particular it is not required that these maps are injective. I was wondering whether there are some conditions on the direct system to ensure that the direct limit object is isomorphic to a direct limit object which is computed using a direct system with only injective structure maps.
Suppose we have a direct system of the shape (taking $I = \mathbb N$) $$ A_0 \to A_1 \to A_2 \to \ldots $$ Then we can pass to the following direct system: As objects we take $A'_i = A_i/\ker(f_{i,i+1})$. As structure maps we use the following maps, given by composing the canonical maps: $$ f'_{i,i+1} \colon A'_i = A_i/\ker(f_{i,i+1}) \to A_{i+1} \to A_{i+1}/\ker(f_{i+1,i+2}) = A'_{i+1} $$ Now if I am not mistaken, we have that $\operatorname{colim}_i A_i \cong \operatorname{colim}_i A'_i$. Of course the new structure maps need not be injective, since we pass to a quotient of $A_{i+1}$. However we can iterate this construction. We would need some "uniform" ascending chain condition to actually obtain a direct system with injective maps after finitely many iterations.
Now these are just some vague ideas. I hope someone can give me a hint or references where this question is addressed.