Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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SOME RESULTS OF MONOMIAL IDEALS ON REGULAR SEQUENCES

  • Naghipour, Reza (Department of Mathematics University of Tabriz and School of Mathematics Institute for Research in Fundamental Sciences (IPM)) ;
  • Vosughian, Somayeh (Institute for Advanced Studies in Basic Sciences)
  • Received : 2020.06.03
  • Accepted : 2020.11.02
  • Published : 2021.05.31
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Abstract

Let R denote a commutative noetherian ring, and let 𝐱 := x1, …, xd be an R-regular sequence. Suppose that 𝖆 denotes a monomial ideal with respect to 𝐱. The first purpose of this article is to show that 𝖆 is irreducible if and only if 𝖆 is a generalized-parametric ideal. Next, it is shown that, for any integer n ≥ 1, (x1, …, xd)n = ⋂P(f), where the intersection (irredundant) is taken over all monomials f = xe11 ⋯ xedd such that deg(f) = n - 1 and P(f) := (xe1+11, ⋯, xed+1d). The second main result of this paper shows that if 𝖖 := (𝐱) is a prime ideal of R which is contained in the Jacobson radical of R and R is 𝖖-adically complete, then 𝖆 is a parameter ideal if and only if 𝖆 is a monomial irreducible ideal and Rad(𝖆) = 𝖖. In addition, if a is generated by monomials m1, …, mr, then Rad(𝖆), the radical of a, is also monomial and Rad(𝖆) = (ω1, …, ωr), where ωi = rad(mi) for all i = 1, …, r.

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Acknowledgement

The authors are deeply grateful to the referee for a very careful reading of the manuscript and many valuable suggestions in improving the quality of the paper. The authors also would like to thank Professor Monireh Sedghi for her reading of the first draft and many helpful suggestions.

References

  1. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.
  2. J. A. Eagon and M. Hochster, R-sequences and indeterminates, Quart. J. Math. Oxford Ser. (2) 25 (1974), 61-71. https://doi.org/10.1093/qmath/25.1.61
  3. W. Heinzer, A. Mirbagheri, J. Ratliff, and K. Shah, Parametric decomposition of monomial ideals. II, J. Algebra 187 (1997), no. 1, 120-149. https://doi.org/10.1006/jabr.1997.6831
  4. W. Heinzer, L. J. Ratliff, Jr., and K. Shah, Parametric decomposition of monomial ideals. I, Houston J. Math. 21 (1995), no. 1, 29-52.
  5. J. Herzog and T. Hibi, Monomial Ideals, Graduate Texts in Mathematics, 260, SpringerVerlag London, Ltd., London, 2011. https://doi.org/10.1007/978-0-85729-106-6
  6. K. Kiyek and J. Stuckrad, Integral closure of monomial ideals on regular sequences, Rev. Mat. Iberoamericana 19 (2003), no. 2, 483-508. https://doi.org/10.4171/RMI/359
  7. M. Nagata, Local Rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York, 1962.
  8. M. Sedghi, Asymptotic behaviour of monomial ideals on regular sequences, Rev. Mat. Iberoam. 22 (2006), no. 3, 955-962. https://doi.org/10.4171/RMI/479
  9. D. K. Taylor, Ideals generated by monomials in an R-sequence, Ph.D. dissertation, University of Chicago, 1966.