  Let $R=\oplus_{i\in \N_0}R_n$ be a standard graded ring, $R_+ :=\oplus_{i\in
\N}R_n$ be the irrelevant ideal of $R$ and $\fa_0$ be an ideal of $R_0$. In
this paper, as a generalization of the concept of Castelnouvo-Mumford
regularity $\reg(M)$ of a finitely generated graded $R$-module $M$, we define
the regularity of $M$ with respect to $\fa_0+ R_+$, say $\reg_{\fa_0+ R_+}(M)$.
And we study some relations of this new invariant with the classic one. To this
end, we need to consider the cohomological dimension of some finitely generated
$R_0$-modules. Also, we will express $\reg_{\fa_0+ R_+}(M)$ in terms of some
invariants of the minimal graded free resolution of $M$ and see that in a
special case this invariant is independent of the choice of $\fa_0$.
