  A pair of ideals $J\subseteq I\subseteq R$ has been called Aluffi
torsion-free if the Aluffi algebra of $I/J$ is isomorphic with the
corresponding Rees algebra. We give necessary and sufficient conditions for the
Aluffi torsion-free property in terms of the first syzygy module of the form
ideal $J^*$ in the associated graded ring of $I$. For two pairs of ideals
$J_1,J_2\subseteq I$ such that $J_1-J_2\in I^2$, we prove that if one pair is
Aluffi torsion-free the other one is so if and only if the first syzygy modules
of $J_1$ and $J_2$ have the same form ideals. We introduce the notion of
strongly Aluffi torsion-free ideals and present some results on these ideals.
