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n-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS

Year 2019, , 204 - 223, 11.07.2019
https://doi.org/10.24330/ieja.587073

Abstract

In a commutative ring R with unity, given an ideal I of R, Anderson and Badawi in 2011 introduced the invariant w(I), which is the minimal
integer n for which I is an n-absorbing ideal of R. In the speci fic case that
R = k[x1;...; xn] is a polynomial ring over a field k in n variables x1; ... ; xn,
we calculate w(I) for certain monomial ideals I of R.

References

  • D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra, 39(5) (2011), 1646-1672.
  • D. F. Anderson and S. T. Chapman, How far is an element from being prime?, J. Algebra Appl., 9(5) (2010), 779-789.
  • A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75(3) (2007), 417-429.
  • A. Badawi, n-Absorbing ideals of commutative rings and recent progress on three conjectures: a survey, In: Fontana, M., Frisch, S., Glaz, S., Tartarone, F., Zanardo, P. (eds) Rings, Polynomials, and Modules, Springer, Cham, (2017), 33-52.
  • J. Chen, S. Morey and A. Sung, The stable set of associated primes of the ideal of a graph, Rocky Mountain J. Math., 32(1) (2002), 71-89.
  • H. S. Choi and A. Walker, The radical of an n-absorbing ideal, arXiv:1610.10077 [math.AC], (2016), J. Commutative Algebra, accepted.
  • G. Donadze, The Anderson-Badawi Conjecture for commutative algebras over in finite fi elds, Indian J. Pure Appl. Math., 47(4) (2016), 691-696.
  • G. Donadze, A proof of the Anderson-Badawi rad(I)^n < I formula for n- absorbing ideals, Proc. Indian Acad. Sci. Math. Sci., 128(1) (2018), Art. 6 (6 pp).
  • D. Eisenbud, Commutative Algebra, with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.
  • J. Herzog and T. Hibi, Monomial Ideals, Graduate Texts in Mathematics, 260, Springer-Verlag London, Ltd., London, 2011.
  • A. Laradji, On n-absorbing rings and ideals, Colloq. Math., 147(2) (2017), 265-273.
  • E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005.
  • H. F. Moghimi and S. R. Naghani, On n-absorbing ideals and the n-Krull dimension of a commutative ring, J. Korean Math. Soc., 53(6) (2016), 1225- 1236.
  • W. F. Moore, M. Rogers and S. Sather-Wagstaff, Monomial Ideals and Their Decompositions, Universitext, Springer, Cham, 2018, www.ndsu.edu/pubweb/ ssatherw/DOCS/monomial.pdf.
  • P. Nasehpour, On the Anderson-Badawi wR[X](I[X]) = wR(I) conjecture, Arch. Math. (Brno), 52(2) (2016), 71-78.
  • V. C. Quinonez, Integral closure and other operations on monomial ideals, J. Commut. Algebra, 2(3) (2010), 359-386.
  • S. Sihem and H. Sana, On Anderson-Badawi conjectures, Beitr. Algebra Geom., 58(4) (2017), 775-785.
  • A. Simis, W. V. Vasconcelos and R. H. Villarreal, On the ideal theory of graphs, J. Algebra, 167 (1994), 389-416.
Year 2019, , 204 - 223, 11.07.2019
https://doi.org/10.24330/ieja.587073

Abstract

References

  • D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra, 39(5) (2011), 1646-1672.
  • D. F. Anderson and S. T. Chapman, How far is an element from being prime?, J. Algebra Appl., 9(5) (2010), 779-789.
  • A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75(3) (2007), 417-429.
  • A. Badawi, n-Absorbing ideals of commutative rings and recent progress on three conjectures: a survey, In: Fontana, M., Frisch, S., Glaz, S., Tartarone, F., Zanardo, P. (eds) Rings, Polynomials, and Modules, Springer, Cham, (2017), 33-52.
  • J. Chen, S. Morey and A. Sung, The stable set of associated primes of the ideal of a graph, Rocky Mountain J. Math., 32(1) (2002), 71-89.
  • H. S. Choi and A. Walker, The radical of an n-absorbing ideal, arXiv:1610.10077 [math.AC], (2016), J. Commutative Algebra, accepted.
  • G. Donadze, The Anderson-Badawi Conjecture for commutative algebras over in finite fi elds, Indian J. Pure Appl. Math., 47(4) (2016), 691-696.
  • G. Donadze, A proof of the Anderson-Badawi rad(I)^n < I formula for n- absorbing ideals, Proc. Indian Acad. Sci. Math. Sci., 128(1) (2018), Art. 6 (6 pp).
  • D. Eisenbud, Commutative Algebra, with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.
  • J. Herzog and T. Hibi, Monomial Ideals, Graduate Texts in Mathematics, 260, Springer-Verlag London, Ltd., London, 2011.
  • A. Laradji, On n-absorbing rings and ideals, Colloq. Math., 147(2) (2017), 265-273.
  • E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005.
  • H. F. Moghimi and S. R. Naghani, On n-absorbing ideals and the n-Krull dimension of a commutative ring, J. Korean Math. Soc., 53(6) (2016), 1225- 1236.
  • W. F. Moore, M. Rogers and S. Sather-Wagstaff, Monomial Ideals and Their Decompositions, Universitext, Springer, Cham, 2018, www.ndsu.edu/pubweb/ ssatherw/DOCS/monomial.pdf.
  • P. Nasehpour, On the Anderson-Badawi wR[X](I[X]) = wR(I) conjecture, Arch. Math. (Brno), 52(2) (2016), 71-78.
  • V. C. Quinonez, Integral closure and other operations on monomial ideals, J. Commut. Algebra, 2(3) (2010), 359-386.
  • S. Sihem and H. Sana, On Anderson-Badawi conjectures, Beitr. Algebra Geom., 58(4) (2017), 775-785.
  • A. Simis, W. V. Vasconcelos and R. H. Villarreal, On the ideal theory of graphs, J. Algebra, 167 (1994), 389-416.
There are 18 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Hyun Seung Choi This is me

Andrew Walker This is me

Publication Date July 11, 2019
Published in Issue Year 2019

Cite

APA Choi, H. S., & Walker, A. (2019). n-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS. International Electronic Journal of Algebra, 26(26), 204-223. https://doi.org/10.24330/ieja.587073
AMA Choi HS, Walker A. n-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS. IEJA. July 2019;26(26):204-223. doi:10.24330/ieja.587073
Chicago Choi, Hyun Seung, and Andrew Walker. “N-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS”. International Electronic Journal of Algebra 26, no. 26 (July 2019): 204-23. https://doi.org/10.24330/ieja.587073.
EndNote Choi HS, Walker A (July 1, 2019) n-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS. International Electronic Journal of Algebra 26 26 204–223.
IEEE H. S. Choi and A. Walker, “n-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS”, IEJA, vol. 26, no. 26, pp. 204–223, 2019, doi: 10.24330/ieja.587073.
ISNAD Choi, Hyun Seung - Walker, Andrew. “N-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS”. International Electronic Journal of Algebra 26/26 (July 2019), 204-223. https://doi.org/10.24330/ieja.587073.
JAMA Choi HS, Walker A. n-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS. IEJA. 2019;26:204–223.
MLA Choi, Hyun Seung and Andrew Walker. “N-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS”. International Electronic Journal of Algebra, vol. 26, no. 26, 2019, pp. 204-23, doi:10.24330/ieja.587073.
Vancouver Choi HS, Walker A. n-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS. IEJA. 2019;26(26):204-23.