  For any vector bundle $V$ on a toric Deligne-Mumford stack $\ix$ the
formalism of \cite{EJK:16} defines two intertial products $\star_{V^{+}}$ and
$\star_{V^{-}}$ on the Chow group of the inertia stack. We give an explicit
presentation for the integral $\star_{V^+}$ and $\star_{V^-}$ Chow rings,
extending earlier work of Boris-Chen-Smith \cite{BCS:05} and Jiang-Tsen
\cite{JiTs:10} in the orbifold Chow ring case, which corresponds to $V = 0$.
  We also describe an {\em asymptotic} product on the rational Chow group of
the inertia stack obtained by letting the rank of the bundle $V$ go to
infinity.
