Exercise from Rotman, direct limit of quotient modules

Question

Suppose $$0 \to U \to V \to V/U \to 0$$ is an exact sequence of left $R$-modules.

Let $\lbrace U_i, \alpha^{i}_j\rbrace$ be a direct sequence of submodules of $U$, where $$\alpha^{i}_j : U_i \to U_j \ \ \ \ i \leq j$$ is the inclusion. Moreover we have $$U = \underrightarrow{\lim} U_i.$$

Then we see that $\lbrace V/U_i, e^{i}_j\rbrace$ with $$e^{i}_j : V/U_i \to V/U_j \ \ \ \ i \leq j$$ $$v + U_i \to v + U_j$$ is a direct system.

I have to show that $\underrightarrow{\lim}V/U_i \cong V/U$.

I don't know how to define a map from $V/U$ to another cocone over $\lbrace V/U_i, e^{i}_j\rbrace$, any hint ?

Answer 1

Hint: Say we have a cocone $\left(V/U_i\overset{g_i}\to W\right)_i$. Then consider the canonical map $\varphi:V\to V/U_i\to W$. By the cocone property, it doesn't depend on $i$, and show that $U\le \ker\varphi$.