n-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS

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.

Keywords

• n-Absorbing ideal,monomial ideal,Noether exponent

References

1. D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra, 39(5) (2011), 1646-1672.
2. D. F. Anderson and S. T. Chapman, How far is an element from being prime?, J. Algebra Appl., 9(5) (2010), 779-789.
3. A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75(3) (2007), 417-429.
4. 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.
5. 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.
6. H. S. Choi and A. Walker, The radical of an n-absorbing ideal, arXiv:1610.10077 [math.AC], (2016), J. Commutative Algebra, accepted.
7. G. Donadze, The Anderson-Badawi Conjecture for commutative algebras over infinite fields, Indian J. Pure Appl. Math., 47(4) (2016), 691-696.
8. 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).

9. D. Eisenbud, Commutative Algebra, with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.
10. J. Herzog and T. Hibi, Monomial Ideals, Graduate Texts in Mathematics, 260, Springer-Verlag London, Ltd., London, 2011.
11. A. Laradji, On n-absorbing rings and ideals, Colloq. Math., 147(2) (2017), 265-273.
12. E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005.
13. 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.
14. 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.
15. P. Nasehpour, On the Anderson-Badawi wR[X](I[X]) = wR(I) conjecture, Arch. Math. (Brno), 52(2) (2016), 71-78.
16. V. C. Quinonez, Integral closure and other operations on monomial ideals, J. Commut. Algebra, 2(3) (2010), 359-386.
17. S. Sihem and H. Sana, On Anderson-Badawi conjectures, Beitr. Algebra Geom., 58(4) (2017), 775-785.
18. A. Simis, W. V. Vasconcelos and R. H. Villarreal, On the ideal theory of graphs, J. Algebra, 167 (1994), 389-416.