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Depth of an ideal on a pair of modules | ||
| Journal of Hyperstructures | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 04 آبان 1404 اصل مقاله (1.59 M) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22098/jhs.2025.17916.1116 | ||
| نویسنده | ||
| Morteza Lotfi Parsa* | ||
| Department of Mathematics, Sayyed Jamaleddin Asadabadi University, Asadabad, Iran | ||
| چکیده | ||
| Let R be a commutative Noetherian ring and I an ideal of R. Suppose that S is a Serre subcategory of the category of R-modules which satisfies the condition CI. Let $M$ be a ZD-module and N an R-module. As a generalization of the notion of S-depth(I, M), we define the S-depth of I on the pair (N, M) by S-depth(I, N, M):=S-depth(AnnR(N/IN), M). We investigate the connections between S-depth(I, N, M), local cohomology modules and Ext functors. In particular, when N is finitely generated, it is shown that S-depth(I, N, M)=inf {i: H{i}I(N, M) ∉ S}=inf{i: ExtiR(N/{IN}, M)∉ S}. Moreover, various formulas are provided that relate this generalized S-depth to other notions of depth in the literature. | ||
| کلیدواژه‌ها | ||
| Depth؛ Local cohomology؛ Serre subcategory؛ ZD-module | ||
| مراجع | ||
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[1] M. Aghapournahr and L. Melkersson, Local cohomology and Serre subcategories, J. Algebra, 320(3) (2008), 1275–1287. [2] M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasg. Math. J., 21(1) (1980), 173–181. [3] L. Chu and Z. Tang, On the Artinianness of local cohomology, Comm. Algebra, 35(12) (2007), 3821–3827. [4] K. Divaani-Aazar and M. A. Esmkhani, Artinianness of local cohomology modules of ZD-modules, Comm. Algebra, 33(8) (2005), 2857–2863. [5] E. G. Evans, Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc., 155(2) (1971), 505–512. [6] A. Fathi, A. Tehranian and H. Zakeri, Filter regular sequences and generalized local cohomology modules, Bull. Malays. Math. Sci. Soc., 38 (2015), 467–482. [7] M. Lotfi Parsa, Depth of an ideal on ZD-modules, Publ. Inst. Math. (Beograd) (N.S.), 120 (2019), 29–37. [8] M. Lotfi Parsa, S- depth on ZD-modules and local cohomology, Czechoslovak Math. J., 71 (2021), 755–764. [9] D. Rees, The grade of an ideal or module, Math. Proc. Cambridge Philos. Soc., 53(1) (1957), 28–42. [10] R. Y. Sharp, Steps in Commutative Algebra, Cambridge Univ. Press, Cambridge, 2000. [11] A. Vahidi and M. Aghapournahr, Some results on generalized local cohomology modules, Comm. Algebra, 43(5) (2015), 2214–2230. | ||
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