Advertisement

Results in Mathematics

, 74:140 | Cite as

Normality of Cover Ideals of Graphs and Normality Under Some Operations

  • Ibrahim Al-AyyoubEmail author
  • Mehrdad Nasernejad
  • Leslie G. Roberts
Article
First Online:
  • 56 Downloads

Abstract

We produce a procedure for constructing new normal monomial ideals from other ideals that are assumed to be normal. This enables us to prove that if the cover ideal of a graph G is normal, then the cover ideal of the graph H is normal as well, where the graph H is obtained by connecting all vertices in G with a new vertex. We use these ideas to explore the normality of the cover ideals of some imperfect graphs. Also, we investigate the normality under expansion, this leads us to generalize the work of Al-Ayyoub (Rocky Mountain J Math 39:1–9, 2009). Furthermore, we investigate the normality under more operations such as weighting, polarization, localization, contraction, and deletion.

Keywords

Normal ideals cover ideals monomial operators imperfect graphs 

Mathematics Subject Classification

13B22 05C25 05E40 
This is a preview of subscription content, log in to check access.

Notes

References

  1. 1.
    Al-Ayyoub, I.: Normality of monomial ideals. Rocky Mountain J. Math. 39, 1–9 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Al-Ayyoub, I., Jaradat, I., Al-Zoubi, K.: On the normality of a class of monomial ideals via the Newton polyhedron. Mediterr. J. Math. 16(3), 1–16 (2019)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bayati, S., Herzog, J.: Expansions of monomial ideals and multigraded modules. Rocky Mountain J. Math. 44, 1781–1804 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bruns, W., Ichim, B., RÖmer, T., Sieg, R., SÖger, C.: Normaliz. Algorithms for Rational Cones and Affine Monoids. https://www.normaliz.uni-osnabrueck.de/home/how-to-cite-normaliz/
  5. 5.
    Coughlin, H.: Classes of Normal Monomial Ideals, Ph.D. thesis. University of Oregon, Eugene, OR (2004)Google Scholar
  6. 6.
    Escobar, C., Villarreal, R.H., Yoshino, Y.: Torsion-Freeness and Normality of Blowup Rings of Monomial Ideals, Commutative Algebra, Lecture Notes in Pure and Applied Mathematics, vol. 244, pp. 69–84. Chapman & Hall/CRC, Boca Raton (2006)zbMATHGoogle Scholar
  7. 7.
    Faridi, S.: Monomial Ideals Via Square-free Monomial Ideals. Lecture Notes in Pure and Applied Mathematics, vol. 244, pp. 85–114. CRC Press, Boca Raton (2005) zbMATHGoogle Scholar
  8. 8.
    Grayson, D.R., Stillman, M.E.: Macaulay2, A Software System for Research in Algebraic Geometry. http://www2.macaulay2.com/Macaulay2/Citing/
  9. 9.
    Hà, H.T., Morey, S.: Embedded associated primes of powers of square-free monomial ideals. J. Pure Appl. Algebra 214, 301–308 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Herzog, J., Hibi, T.: Monomial Ideals, Graduate Texts in Mathematics, vol. 260. Springer, Berlin (2011)CrossRefGoogle Scholar
  11. 11.
    Herzog, J., Qureshi, A.A.: Persistence and stability properties of powers of ideals. J. Pure Appl. Algebra 219, 530–542 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ichim, B., KatthÄn, L., Moyano-FernÁndez, J.J.: The behavior of Stanley depth under polarization. J. Comb. Theory Ser. A 135, 332–347 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jahan, A.S.: Prime filteration of monomila ideals under polarization. J. Algebra 312, 1011–1032 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kaiser, T., Stehlík, M., Škrekovski, R.: Replication in critical graphs and the persistence of monomial ideals. J. Comb. Theory Ser. A 123, 239–251 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Martnez-Bernal, J., Morey, S., Villarreal, R.H.: Associated primes of powers of edge ideals. Collect. Math. 63, 361–374 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Martnez-Bernal, J., Morey, S., Villarreal, R.H., Vivares, C.E.: Depth and regularity of monomial ideals via polarization and combinatorial optimization. Acta Mathematica Vietnamica 44(1), 243–268 (2019)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Nasernejad, M.: Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications. J. Algebra Relat. Top. 2, 15–25 (2014)zbMATHGoogle Scholar
  18. 18.
    Nasernejad, M.: Persistence property for some classes of monomial ideals of a polynomial ring. J. Algebra Appl. 16(5), 1750105 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nasernejad, M., Khashyarmanesh, K.: On the Alexander dual of the path ideals of rooted and unrooted trees. Comm. Algebra 45, 1853–1864 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nasernejad, M., Khashyarmanesh, K., Al-Ayyoub, I.: Associated primes of powers of cover ideals under graph operations. Comm. Algebra (2019).  https://doi.org/10.1080/00927872.2018.1527920 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rajaee, S., Nasernejad, M., Al-Ayyoub, I.: Superficial ideals for monomial ideals. J. Algebra Appl. 16(2), 1850102 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Reid, L., Roberts, L.G., Vitulli, M.A.: Some results on normal monomial ideals. Comm. Algebra 31, 4485–4506 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Restuccia, G., Villarreal, R.H.: On the normality of monomial ideals of mixed products. Comm. Algebra 29, 3571–3580 (2001)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sayedsadeghi, M., Nasernejad, M.: Normally torsion-freeness of monomial ideals under monomial operators. Comm. Algebra 46, 5447–5459 (2018) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Simis, A., Vasconcelos, W., Villarreal, R.: On the ideal theory of graphs. J. Algebra 167, 389–416 (1994)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Swanson, I., Huneke, C.: Integral Closure of Ideals, Rings, and Modules. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  27. 27.
    Vasconcelos, W.V.: Arithmetic of Blowup Algebras. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  28. 28.
    Villarreal, R.H.: Monomial Algebras. Monographs and Research Notes in Mathematics, 2nd edn. CRC Press, Boca Raton (2015)Google Scholar
  29. 29.
    Villarreal, R.H.: Normality of subrings generated by square-free monomials. J. Pure Appl. Algebra 113, 91–106 (1996)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Villarreal, R.H.: Rees cones and monomial rings of matroids. Linear Algebra Appl. 428, 2933–2940 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsJordan University of Science and TechnologyIrbidJordan
  2. 2.Department of MathematicsKhayyam UniversityMashhadIran
  3. 3.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

Personalised recommendations