admissible ideals,, bound quiver and its algebra

Assume, that $Q$ is a quiver and $k$ is a field. Let $kQ$ be the associated path algebra. Denote by $R_Q$ the two-sided ideal in $kQ$ generated by all paths of length $1$, i.e. all arrows. This ideal is known as the arrow ideal.

It is easy to see, that for any $m⩾1$ we have that $R_Q^m$ is a two-sided ideal generated by all paths of length $m$. Note, that we have the following chain of ideals:

$$R_Q^2\supseteq R_Q^3\supseteq R_Q^4\supseteq \mathrm{\cdots }$$

Definition. A two-sided ideal $I$ in $kQ$ is said to be admissible if there exists $m⩾2$ such that

$$R_Q^m\subseteq I\subseteq R_Q^2.$$

If $I$ is an admissible ideal in $kQ$, then the pair $(Q,I)$ is said to be a bound quiver and the quotient algebra $kQ/I$ is called bound quiver algebra.

The idea behind this is to treat some paths in a quiver as equivalent. For example consider the following quiver

Then the ideal generated by $ab-c$ is not admissible ($ab-c\notin R_Q^2$) but an ideal generated by $ab-ef$ is. We can see that this means that ,,walking” from $1$ to $3$ directly and through $2$ is not the same, but walking in the same number of steps is.

Note, that in our case there is no path of length greater then $2$. In particular, for any $m>2$ we have $R_Q^m=0$.

More generally, it can be easily checked, that if $Q$ is a finite quiver without oriented cycles, then there exists $m\in \mathbb{N}$ such that $R_Q^m=0$