Results on the normality of square-free monomial ideals and cover ideals under some graph operations


  • Ibrahim Al-Ayyoub
  • Mehrdad Nasernejad
  • Kazem Khashyarmanesh
  • Leslie G. Roberts
  • Veronica Crispin Quiñonez



In this paper, we introduce techniques for producing normal square-free monomial ideals from old such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations. Square-free monomial ideals that come out as linear combinations of two normal ideals are shown to be not necessarily normal; under such a case we investigate the integral closedness of all powers of these ideals.


Al-Ayyoub, I., Nasernejad, M., and Roberts, L., Normality of Monomial Ideals under some Operations with Application to Cover Ideals, Results Math. 74 (2019), no. 4, Paper No. 140, 26 pp.

Bondy, J. A., and Murty, U. S. R., Graph theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. xii+651 pp.

Bruns, W., and Horn, M., Normaliz 3.8.4 and various earlier versions 3.0.0 and later with different authors. Documentation included with the program Normaliz: Algorithms for rational cones and affine monoids.

Available at

Escobar, C., Villarreal, R. H., and Yoshino, Y., Torsion-freeness and normality of blowup rings of monomial ideals, Commutative Algebra, 69–84, Lect. Notes Pure Appl. Math. 244, Chapman & Hall/CRC, Boca Raton, FL, 2006.

Faridi, S., Monomial Ideals via Square-free Monomial Ideals, Commutative Algebra, 85–114, Lecture Notes in Pure and Applied Math. 244, Chapman & Hall/CRC, Boca Raton, FL, 2006.

Herzog, J., and Hibi, T., Monomial Ideals, Graduate Texts in Mathematics 260 Springer-Verlag London, Ltd., London, 2011.

Kaiser, T., Stehlík, M., and Škrekovski, R., Replication in critical graphs and the persistence of monomial ideals, J. Combin. Theory Ser. A 123 (2014), 239–251.

Khashyarmanesh K., and Nasernejad, M., On the minimal path cover sets of union of certain graphs, Quaest. Math. 43 (2020), no. 7, 881–901.

Martínez-Bernal, J., Morey, S., and Villarreal, R. H., Associated primes of powers of edge ideals, Collect. Math. 63 (2012), no. 3, 361–374.

Nasernejad, M., Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications, J. Algebra Relat. Top. 2 (2014) 15–25.

Nasernejad, M., Persistence property for some classes of monomial ideals of a polynomial ring, J. Algebra Appl. 16, (2017), no. 6, 1750105, 17 pp.

bibitem N3Nasernejad, M., and Khashyarmanesh, K., On the Alexander dual of the path ideals of rooted and unrooted trees, Comm. Algebra 45 (2017), no. 5, 1853–1864.

Nasernejad, M., and Khashyarmanesh, K., Sets defining minimal vertex covers of uniform hypergraphs, Ars Combin. 147 (2019) 135–142.

Nasernejad, M., Khashyarmanesh, K., and Al-Ayyoub, I., Associated primes of powers of cover ideals under graph operations, Comm. Algebra, 47 (2019), no. 5, 1985–1996.

Rajaee, S., Nasernejad, M., and Al-Ayyoub, I., Superficial ideals for monomial ideals, J. Algebra Appl. 16 (2018), no. 6, 1850102, 28 pp.

Restuccia. G., and Villarreal, R. H., On the normality of monomial ideals of mixed products, Comm. Algebra 29 (2001), no. 8, 3571–3580.

Sharifan, L., Nasernejad, M., and Khashyarmanesh, K., Minimal path cover sets and monomial ideals, J. Algebra Appl. 14 (2015), no. 2, 1550014, 14 pp.

Simis, A., Vasconcelos, W., and Villarreal, R. H., On the ideal theory of graphs, J. Algebra 167 (1994), no. 2, 389–416.

Villarreal, R. H., Monomial algebras, Second edition. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015.

Villarreal, R. H., Normality of subrings generated by square-free monomials, J. Pure Appl. Algebra 113 (1996), no. 1, 91–106.

Villarreal, R. H., Rees cones and monomial rings of matroids, Linear Algebra Appl. 428 (2008), no. 11–12, 2933–2940.

Villarreal, R. H., Rees algebras and polyhedral cones of ideals of vertex covers of perfect graphs, J. Algebraic Combin. 27 (2008), no. 6, 293–305.



How to Cite

Al-Ayyoub, I., Nasernejad, M., Khashyarmanesh, K., Roberts, L. G., & Quiñonez, V. C. (2021). Results on the normality of square-free monomial ideals and cover ideals under some graph operations. MATHEMATICA SCANDINAVICA, 127(3).