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 $IM \neq M$.
Ahmadi Amoli, K., Habibi, Z., & Behboodi, R. (2023). Some applications of $k$-regular sequences and arithmetic rank of an ideal with respect to modules. Journal of Algebra and Related Topics, 11(2), 21-35. doi: 10.22124/jart.2023.24393.1518
MLA
Kh. Ahmadi Amoli; Z. Habibi; R. Behboodi. "Some applications of $k$-regular sequences and arithmetic rank of an ideal with respect to modules". Journal of Algebra and Related Topics, 11, 2, 2023, 21-35. doi: 10.22124/jart.2023.24393.1518
HARVARD
Ahmadi Amoli, K., Habibi, Z., Behboodi, R. (2023). 'Some applications of $k$-regular sequences and arithmetic rank of an ideal with respect to modules', Journal of Algebra and Related Topics, 11(2), pp. 21-35. doi: 10.22124/jart.2023.24393.1518
VANCOUVER
Ahmadi Amoli, K., Habibi, Z., Behboodi, R. Some applications of $k$-regular sequences and arithmetic rank of an ideal with respect to modules. Journal of Algebra and Related Topics, 2023; 11(2): 21-35. doi: 10.22124/jart.2023.24393.1518
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