Some applications of $k$-regular sequences and arithmetic rank of an ideal with respect to modules

‎Let $R$ be a commutative Noetherian ring with identity‎, ‎$I$ be an ideal of $R$‎, ‎and $M$ be an $R$-module‎. ‎Let $k\geqslant‎ -‎1$ be an arbitrary integer‎. ‎In this paper‎, ‎we introduce the notions of $\Rad_M(I)$ and $\ara_M(I)$‎ ‎as the radical and the arithmetic rank of $I$ with respect to $M$‎, ‎respectively‎. ‎We show that the existence of some sort of regular sequences‎ ‎can be depended on $\dim M/IM$ and so‎, ‎we can get some information about local cohomology modules as well‎. ‎Indeed‎, ‎if $\ara_M(I)=n\geq 1$ and ${(\Supp_{R}(M/IM))}_{>k}=\emptyset$‎ ‎(e.g.‎, ‎if $\dim M/IM=k$)‎, ‎then there exist $n$ elements $x_1‎, ..., ‎x_n$ in $I$‎ ‎which is a poor $k$-regular $M$-sequence and generate an ideal‎ ‎with the same radical as $\Rad_M(I)$ and so‎ ‎$H^i_I(M)\cong H^i_{(x_1‎, ..., ‎x_n)}(M)$ for all $i\in \mathbb{N}_0$‎. ‎As an application‎, ‎we show that $\ara_M(I) \leq \dim M+1$‎, ‎which is a refinement of the inequality $\ara_R(I) \leq \dim R+1$ for modules‎, ‎attributed to Kronecker and Forster‎. ‎Then‎, ‎we prove‎ ‎$\dim M-\dim M/IM \leq \cd(I‎, ‎M) \leq \ara_M(I) \leq \dim M$‎, ‎if $(R‎, ‎\mathfrak{m})$ is a local ring and $IM \neq M$‎.