  Fortin and Reutenauer defined the non-commutative rank for a matrix with
entries that are linear functions. The non-commutative rank is related to
stability in invariant theory, non-commutative arithmetic circuits, and
Edmonds' problem. We will generalize the non-commutative rank to the
representation theory of quivers and define non-commutative Hom and Ext spaces.
We will relate these new notions to King's criterion for $\sigma$-stability of
quiver representations, and the general Hom and Ext spaces studied by
Schofield. We discuss polynomial time algorithms that compute the
non-commutative Homs and Exts and find an optimal witness for the
$\sigma$-semi-stability of a quiver representation.
