Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 2 / 4 / i / Solution

Take the quiver with one loop. Its path algebra has basis $1,a,a^2,\dots$ and multiplication $a^r{a}^s=a^{r+s}$, so it is the polynomial ring $k[a]$. HenceThe algebra generators need not form a spanning vector-space set: products produce the infinitely many independent paths.