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This quiver has one vertex and $r$ loops. Paths are words in the loops, so its path algebra is a free associative algebra. A representation assigns an arbitrary endomorphism to each loop.

An indecomposable finite-dimensional quiver representation that is not a brick module contains a brick with nonzero self-extensions. The Ringel form then gives $q_Q(\dim B)=1-\dim {\mathrm{Ext}}_Q^1(B,B)\le 0$, impossible for a positive definite Tits form of a quiver.

A vertex of a quiver is a sink when no arrow has that vertex as its source. Dually, a source is a vertex at which no arrow ends.

Set $A=kQ$, $P_i=A{e}_i$, $X_i=e_{i}X$. There is an exact sequencewhere $d(p\otimes x)=p\rho \otimes x-p\otimes f_{\rho }(x)$ in the source and target summands, and the augmentation is $p\otimes x\mapsto px$. Each $P_i$ is a summand of $A$, so the two terms are projective modules. This also works for infinite-dimensional modules.

This quiver has $r$ source vertices, each with one arrow to a common sink. Its path algebra has the vertex idempotents and the arrows as basis. After extracting source kernels from a representation, its remaining arrows identify the source spaces with subspaces of the sink.