  Let $R$ be a commutative Noetherian ring and let $n$ be a non-negative
integer. In this article, by using the theory of Gorenstein dimensions, it is
shown that whenever $R$ is a homomorphic image of a Noetherian Gorenstein ring,
then the invariants $\inf\{i\in\nat_0|\, {\dim\Supp}(\fb^tH_{\fa}^i(M))\geq
n\text{for all} t\in\nat_0\}$ and $\inf\{\lambda_{\fa R_{\p}}^{\fb
R_{\p}}(M_{\p})|\,\p\in {\rm Spec} \, R \text{and} \dim R/ \p\geq n\}$ are
equal, for every finitely generated $R$-module $M$ and for every ideals $\frak
a, \frak b$ of $R$ with $\frak b\subseteq \frak a$. This generalizes the
Faltings' Annihilator Theorem [G. Faltings, {\it \"Uber die Annulatoren lokaler
Kohomologiegruppen}, Arch. Math. {\bf30} (1978) 473-476].
