First Isomorphism Theorem for quivers
Let $Q=({Q}_{0},{Q}_{1},s,t)$ and ${Q}^{\prime}=({Q}_{0}^{\prime},{Q}_{1}^{\prime},{s}^{\prime},{t}^{\prime})$ be quivers. Assume, that $F:Q\to {Q}^{\prime}$ is a morphism of quivers. Define an equivalence relation^{} $\sim $ on $Q$ as follows: for any $a,b\in {Q}_{0}$ and any $\alpha ,\beta \in {Q}_{1}$ we have
$$a{\sim}_{0}b\text{if and only if}{F}_{0}(a)={F}_{0}(b);$$ 
$$\alpha {\sim}_{1}\beta \text{if and only if}{F}_{1}(\alpha )={F}_{1}(\beta ).$$ 
It can be easily checked that $\sim =({\sim}_{0},{\sim}_{1})$ is an equivalence relation on $Q$.
Using standard techniques we can prove the following:
First Isomorphism Theorem for quivers. The mapping
$$\overline{F}:(Q/\sim )\to \mathrm{Im}(F)$$ 
(where on the left side we have the quotient quiver (http://planetmath.org/QuotientQuiver) and on the right side the image of a quiver (http://planetmath.org/SubquiverAndImageOfAQuiver)) given by
$${\overline{F}}_{0}([a])={F}_{0}(a),{\overline{F}}_{1}([\alpha ])={F}_{1}(\alpha )$$ 
is an isomorphism^{} of quivers.
Proof. It easily follows from the definition of $\sim $ that $\overline{F}$ is a welldefined morphism of quivers. Thus it is enough to show, that $\overline{F}$ is both ,,onto” and ,,11” (in the sense that corresponding components^{} of $\overline{F}$ are).

1.
We will show, that $\overline{F}$ is onto, i.e. both ${\overline{F}}_{0},{\overline{F}}_{1}$ are onto. Let $b\in \mathrm{Im}{(F)}_{0}$ and $\beta \in \mathrm{Im}{(F)}_{1}$. By definition
$${F}_{0}(a)=b,{F}_{1}(\alpha )=\beta $$ for some $a\in {Q}_{0}$, $\alpha \in {Q}_{1}$. It follows that
$${\overline{F}}_{0}([a])=b,{\overline{F}}_{1}([\alpha ])=\beta .$$ which completes^{} this part.

2.
$\overline{F}$ is injective^{}. Indeed, if
$${\overline{F}}_{0}([a])={\overline{F}}_{0}([b])$$ then ${F}_{0}(a)={F}_{0}(b)$. But then $a{\sim}_{0}b$ and thus $[a]=[b]$. Analogously we prove the statement for ${\overline{F}}_{1}$.
This completes the proof. $\mathrm{\square}$
Title  First Isomorphism Theorem for quivers 

Canonical name  FirstIsomorphismTheoremForQuivers 
Date of creation  20130322 19:17:25 
Last modified on  20130322 19:17:25 
Owner  joking (16130) 
Last modified by  joking (16130) 
Numerical id  5 
Author  joking (16130) 
Entry type  Definition 
Classification  msc 14L24 