= Proof of Ringel lemma on bricks
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 $I=\operatorname{im}f\subset K=\ker f=\bigoplus_jK_j$. Choose a nonzero component $u:I\to K_j$. The square-zero endomorphism $X\xrightarrow fI\xrightarrow uK_j\hookrightarrow X$ has rank at least $\dim I$, so $u$ is injective.
If $\operatorname{Ext}^1(I,K_j)=0$, 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 $\operatorname{Ext}^1(K_j,K_j)\twoheadrightarrow\operatorname{Ext}^1(I,K_j)$. 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 https://www.math.uni-bielefeld.de/~wcrawley/quivlecs.pdf[William Crawley-Boevey's quiver lectures].
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