arrows in the injective representations of quivers

Question

Let $Q$ be a quiver (a directed graph) and $E$ be a representation of $Q$ by R-modules, that is, for each vertex $v$ in $Q$ there exists an R-module $E(v)$ and for each arrow $a:v\to w$ there exists a module homomorphism $E(v)\to E(w)$. In other words $E$ is a functor from $Q$ to the category of $R$-modules.

Let Rep(Q, R) denotes the category of all representations of $Q$ by R-modules. It is proved that if a representation $E$ in this category be injective then it satisfies in the followng axioms.

1) E(v) is an injective R-module for each vertex $v\in Q$.

2) the morphism $E(v)\to \cap_{Q(v,w)}E(w)$ induced by morphisms $E(v)\to E(w)$ is a split epimorphism. Notice that by $Q(v,w)$ we mean the set of all arrows from $v$ to $w$.

Question: By the properties of representations of quivers can we deduce that each morphism $E(v)\to E(w)$ is also a split epimorphism?

Answer 1

Yes.

Firstly, here's a construction of representations $M_u$ that clearly have the property that the maps $M_u(v)\to M_u(w)$ associated to arrows are split epimorphisms. Let $M$ be an $R$-module, and $u$ a vertex. Then $M_u(v)=\prod_{p:v\leadsto u}M_p$ is a direct product of copies of $M$, indexed by the set of paths $p$ from $v$ to $u$, where, for each arrow $\alpha$ from $v$ to $w$, the associated map $M_u(v)\to M_u(w)$ is zero on the factors $M_p$ indexed by paths $p$ that don't start with $\alpha$, and the identity map $M_p\to M_{p'}$ if $p$ is the concatenation of $\alpha$ and a path $p'$ from $w$ to $u$.

For any representation $E$ and $R$-module homomorphism $\varphi:E(u)\to M$, there is a corresponding homomorphism of representations $E\to M_u$ for which the component of $E(v)\to M_u(v)$ to the factor indexed by a path $p$ is just the composition of the map $E(v)\to E(u)$ induced by $p$ in the representation, and the map $\varphi$. (In fact $M\mapsto M_u$ is right adjoint to the functor $E\mapsto E(u)$ from representations to $R$-modules.)

So there's an embedding of $E$ into $\prod_uE(u)_u$ induced by the identity maps $\{\operatorname{id}_{E(u)}: u\in V(Q)\}$, which splits if $E$ is injective.

Since the property that $E(v)\to E(w)$ is a split epimorphism is clearly preserved by direct products and direct factors, every injective representation has this property.