University of Guilan Journal of Algebra and Related Topics 2345-3931 11 2 2023 12 01 Some applications of $k$-regular sequences and arithmetic rank of an ideal with respect to modules 21 35 7136 10.22124/jart.2023.24393.1518 EN Kh. Ahmadi Amoli Department of Mathematics, Payame Noor University, Tehran, Iran Z. Habibi Department of Mathematics, Payame Noor University, Tehran, Iran R. Behboodi Department of Mathematics, Payame Noor University, Tehran, Iran Journal Article 2023 04 25 ‎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 $IM \neq M$‎. https://jart.guilan.ac.ir/article_7136_fc26de8f20278b85efd2f88a1bd48f02.pdf