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Applying the contravariant Hom functor ${\mathrm{Hom}}_R(-,W)$ to $0\to I\to X\to C\to 0$ givesThe connecting map sends $I\to W$ to its pushout of a module extension. Its vanishing means that map extends to $X\to W$. For a hereditary ring, ${\mathrm{Ext}}^2(C,W)=0$, making the indicated restriction on first extension groups surjective. The covariant variable has its corresponding long exact sequence.

The standard projective resolution of a quiver representation has length one, even for quivers with oriented cycles. Thus every path algebra is a hereditary ring, and higher extension groups vanish. Applying a long exact sequence of Ext groups to $0\to I\to K\to C\to 0$ gives a surjection ${\mathrm{Ext}}^1(K,W)\twoheadrightarrow {\mathrm{Ext}}^1(I,W)$, since the next term ${\mathrm{Ext}}^2(C,W)$ is zero. This is the hereditary step in the Ringel lemma on bricks.