relations in quiver

Let $Q$ be a quiver and $k$ a field.

Definition. A relation in $Q$ is a linear combination (over $k$) of paths of length at least $2$ such that all paths have the same source and target. Thus a relation is an element of the path algebra $kQ$ of the form

$$\rho =\sum _{i=1}^m{\lambda }_i\cdot w_i$$

such that there exist $x,y\in Q_0$ with $s(w_i)=x$ and $t(w_i)=y$ for all $i$, all $w_i$ are of length at least $2$ and not all ${\lambda }_i$ are zero.

If a relation $\rho$ is of the form $\rho =w$ for some path $w$, then it is called a zero relation and if $\rho =w_1-w_2$ for some paths $w_1,w_2$, then $\rho$ is called a commutativity relation.