Ringel lemma on bricks

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.

The Fitting lemma supplies a nonzero nilpotent endomorphism $f$ of a non-brick indecomposable $X$. Minimize its nonzero rank; nilpotence and minimality give $f^2=0$. Set $I=\mathrm{im}f\subset K=\ker f={⨁}_j{K}_j$. Choose a nonzero component $u:I\to K_j$. The square-zero endomorphism $X\stackrel{f}{\to }I\stackrel{u}{\to }{K}_j\hookrightarrow X$ has rank at least $\dim I$, so $u$ is injective.

If ${\mathrm{Ext}}^1(I,K_j)=0$, the projection $K\to K_j$ extends to $X\to K_j$ by the long exact sequence of Ext groups, giving a module retraction and contradicting indecomposability. Thus this extension group is nonzero. Since path algebras are hereditary, $u$ induces a surjection ${\mathrm{Ext}}^1(K_j,K_j)\twoheadrightarrow {\mathrm{Ext}}^1(I,K_j)$. Hence $K_j$ is a proper indecomposable submodule with self-extensions. Iterate until a brick module is reached; dimensions strictly decrease.

This minimal-rank argument is given in section 2 of William Crawley-Boevey's quiver lectures.