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Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$ a non-nilpotentideal of $R$. The purpose of this article is to investigate Faltings'finiteness dimension $f_I(R)$ and equidimensionalness of certain homomorphicimage 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 thering $R/ I+\cup_{n\geq 1}(0:_RI^n)$ is equidimensional of dimension $\dim R-1$.Moreover, we will obtain a lower bound for injective dimension of the localcohomology module $H^{{\rm ht}\ I}_I(R)$, in the case $(R, \frak m)$ is acomplete equidimensional local ring.

Faltings' finiteness dimension of local cohomology modules over local Cohen-Macaulay rings