Abstract
Let R be a standard graded algebra over an infinite field \(\mathbb {K}\) and M a finitely generated \({\mathbb Z}\)-graded R-module. Let \(I_1,\ldots I_m\) be graded ideals of R. The functions \(r(M/I_1^{a_1}\ldots I_m^{a_m}M)\) and \(r(I_1^{a_1}\ldots I_m^{a_m}M)\) are investigated and their asymptotical behaviours are given. Here, \(r(\bullet )\) stands for the reduction number of a finitely generated graded R-module \(\bullet \).
Similar content being viewed by others
References
Bruns, W., Conca, A.: A remark of regularity of powers and products of ideals. J. Pure Appl. Algebra 221, 2801–2808 (2017)
Bruns, W., Herzog, J.: Cohen–Macaulay rings. Cambridges Studies in Advanced Mathematics, 39 (1993)
Cutkosky, S.D., Herzog, J., Trung, N.V.: Asymptotic behaviour of the Castelnuovo–Mumford regularity. Compos. Math. 118, 243–261 (1999)
Herzog, J., Hibi, T.: Monomial Ideals, Graduate Text in Mathematics. Springer, New York (2011)
Lu, D.C.: On the asymptotic linearity of reduction number. J. Algebra 504, 1–9 (2018)
Kodiyalam, V.: Asymptotic behaviour of Castelnuovo–Mumford regularity. Proc. AMS 128, 407–411 (2000)
Trung, N.V., Wang, H.J.: On the asymptotic linearity of Castelnuovo–Mumford regularity. J. Pure Appl. Algebra 201, 42–48 (2005)
Acknowledgements
We would like to express our thanks to the referee for his/her careful reading and good advice. This project is supported by NSFC (No. 11971338) and NSF of Shanghai (No. 19ZR14241000).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mohammad Taghi Dibaei.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lu, D., Wu, T. On Reduction Numbers of Products of Ideals. Bull. Iran. Math. Soc. 46, 1027–1033 (2020). https://doi.org/10.1007/s41980-019-00308-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41980-019-00308-1