Zero direct limit of nonzero objects

Question

Can anyone present to me kindly a directed set of nonzero objects with the zero direct limit?

I first tried $$F(U)=\{f:U \to R \mid f\text{ is continuous}\}$$ in p.507 of "Advanced Modern Algebra" of Rotman, but I realized that it is not the case. Thanks in advance.

Answer 1

The easiest example is a chain of zero maps: $$M \stackrel{0}{\to} M \stackrel{0}{\to} M \stackrel{0}{\to} M \to \cdots$$ A slightly less trivial example is to multiply by a nilpotent element, e.g. $$\mathbb{Z} / 4\mathbb{Z} \stackrel{2}{\to} \mathbb{Z} / 4\mathbb{Z} \stackrel{2}{\to} \mathbb{Z} / 4\mathbb{Z} \stackrel{2}{\to} \mathbb{Z} / 4\mathbb{Z} \to \cdots$$ where the maps are just $x \mapsto 2 x$.

Answer 2

Take your index set $I = \mathbb{N}$, and $X_i$ for all $i \in I$ some arbitrary module with homomorphisms $f_{i,i+1}: X_i \rightarrow X_{i+1}$ such that: $\forall i \in I: \exists j \geq i: f_{j,j+1}=0$