The Commuting Algebra

Let $KQ$ be a path algebra, where $Q$ is a finite quiver and $K$ is a field. We study $KQ/C$ where $C$ is the two-sided ideal in $KQ$ generated by all differences of parallel paths in $Q$. We show that $KQ/C$ is always finite dimensional and its global dimension is finite. Furthermore, we prove that $KQ/C$ is Morita equivalent to an incidence algebra. The paper starts with the more general setting, where $KQ$ is replaced by $KQ/I$ with $I$ a two-sided ideal in $KQ$.