Does the following property of the direct limit of a direct system follow from the axioms for a direct limit?

Solution 1:

A direct limit of a system $(M_i, \mu_{ij})$ is an appropriate family of objects satisfying the universal property. Here Atiyah and Macdonald have constructed a $(M, \mu_i)$ which does the job. It seems like you're worried that a certain property of this entity might come from the particular construction given.

But if $N$ and $\nu_i\colon M_i \to N$ do the job just as well, then there is a (unique) isomorphism $\alpha\colon M \to N$ such that $\alpha \circ \mu_i = \nu_i$ for all $i$. If $\nu_i(x_i) = 0$ then $\alpha(\mu_i(x_i)) = 0$, and hence $\mu_i(x_i) = 0$ because $\alpha$ is an isomorphism. So you are back to exercise 15.

Solution 2:

It might be clearer to say that colimits are characterized in a certain way, that (as usual with categorical things) proves uniqueness up to unique isomorphism. Existence is often proven by giving a construction, indeed. The kind of construction you gave succeeds in any category with coproducts, producing the colimit as a quotient. The property that vanishing in the colimit implies vanishing somewhere along the way ("in finite time"?) does hold in categories of modules: any relation in the colimit involves only finitely-many things, which appear in finite time.

But this property cannot be completely general, because it definitely fails in categories of topological vector spaces, where that quotient must be by the closure of all the relations, in order for the quotient to be Hausdorff.