SOME APPLICATIONS OF k-REGULAR SEQUENCES AND ARITHMETIC RANK OF AN IDEAL WITH RESPECT 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.