  Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$ a non-nilpotent
ideal of $R$. The purpose of this article is to investigate Faltings'
finiteness dimension $f_I(R)$ and equidimensionalness of certain homomorphic
image of $R$. As a consequence we deduce that $f_I(R)={\rm max}\{1, {\rm ht}\
I\}$ and if ${\frak m}\mathrm{Ass}_R(R/I)$ is cotained in Ass$_R(R)$, then the
ring $R/ I+\cup_{n\geq 1}(0:_RI^n)$ is equidimensional of dimension $\dim R-1$.
Moreover, we will obtain a lower bound for injective dimension of the local
cohomology module $H^{{\rm ht}\ I}_I(R)$, in the case $(R, \frak m)$ is a
complete equidimensional local ring.
