I study representations of the quiver Q with two verticies and one arrow (from the 1. to the 2. vertex). In particular, I'm interested to determine wheter or not the representation $V:=(k \overset{id}{\to} k)$ ($k$ is the groundfield over which we consider the representations) is injective or not. Here injectivity means the following: For any injective morphism of representations $j : M \to N$ of $Q$ and any morphism $p : M \to V$ there exists a morphism $g : N \to V$ such that $g \circ j=p$. In other words, for any given diagram of the type
where all spaces are $k$-vectorspaces, all maps are linear, $nj_1=j_2m, p_1=p_2m$ and $j_1,j_2$ are injective, we should be able to construct $g_1,g_2$ linear such that the remaining part of the diagram commutes. I believe that this $V$ is injective (because, well, I cannot find a counter-example, and I have been able to find such for other non-injective representations of $Q$). I thought I was able to prove that $V$ actually is injective, but I've come to doubt my-self. My argument goes like this: We can pick at compliment of $j_1(M_1)$ in $N_1$, say $N_1'$ and declare $g_1(j_1(m_1))=p_1(m_1)$ for all $m_1 \in M_1$ and $g_1(n_1')=0$ for all $n_1' \in N_1'$. Injectivity of $j_1$ makes $g_1$ well-defined and yields $p_1=g_1 \circ j_1$. I thought about defining $g_2$ completely analogous (simply replacing 1 with 2 in the above construction so to say), and while this gives $p_2=g_2j_2$, I'm not sure that $g_2n=g_1$, which is required.
This would be the case if $n(N_1') \subset N_2'$ (!), because in that case, for $n_1=j_1(m_1)+n_1' \in N_1$ with $n_1' \in N_1'$, we would have $n(n_1)=j_2m(m_1)+n(n_1')$ with $n(n_1') \in N_2'$, so $g_2n(n_1)=p_2m(m_1)=p_1(m_1)=g_1(j_1(m_1)+n_1')=g_1(n_1)$. But I cannot figure out if condition (!) holds in general: We know that $n(j_1(M_1)) \subset j_2(M_2)$ by commutativity of the diagram, and that $N_i=j_i(M_i) \oplus N_i'$ for $i=1,2$, but I cannot seem to relate this to (!).
Then I tried replace $N_1,M_1$ with the quotients $N_1/\ker n, M_2/\ker m$ and $N_2,M_2$ with $\text{im}(n), \text{im}(m)$. Because of commutativity of the diagram (and injectivity of $j_2$) I think we can pass all maps to these quotients (and restrict $j_2$ to $\text{im}(m)$, which maps into $\text{im}(n)$). Then I was able to define $g_1,g_2$ with the desired properties in this new diagram, but as above my problem is that I don't know if the maps respect these decompositions that I've made, and so fail to see that $V$ is injective.
Any hints or guidance as to how to precede with this problem would be much appreciated, as I've been stuck for quite a while now. Thank you in advance :)