Depth of an ideal on a pair of modules

Document Type : Research Paper

Author

Department of Mathematics, Sayyed Jamaleddin Asadabadi University, Asadabad, Iran

Abstract

‎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.

Keywords

Main Subjects

References

[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.
Journal of Hyperstructures
Volume 14, Issue 2
December 2025
Pages 184-190

Files

Share

How to cite

Statistics