some applications of k-regular sequencesand arithmetic rank of an ideal withrespect to modules

Let r be a commutative noetherian ring with iden tity, i be an ideal of r, and m be an r-module. let k ⩾ −1 be an arbitrary integer. in this paper, we introduce the notions of radm(i) and aram(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 aram(i) = n ≥ 1 and (suppr(m/im))>k = ∅ (e.g., if dim m/im = k), then there exist n elements x1, ..., xn in i which is a poor k-regular m-sequence and generate an ideal with the same radical as radm(i) and so hi i (m) ∼= hi (x1,...,xn) (m) for all i ∈ n0. as an application, we show that aram(i) ≤ dim m + 1, which is a refinement of the inequality arar(i) ≤ dim r + 1 for modules, attributed to kronecker and forster. then, we prove dim m − dim m/im ≤ cd(i, m) ≤ aram(i) ≤ dim m, if (r, m) is a local ring and im ≠ m.