For every even integer n=2k, let q_n be the ideal <x^{2n},y^{2n},(xy+z^2)^n,z^n> in the polynomial ring R=Q[x,y,z]. In her master's thesis [Y], Yao gives a Groebner basis G for q_n and proves that q_n+I_n contain in (q_n:m), where m is the maximal ideal <x,y,z> of R and I_n is the monomial ideal (x^k)(y^k)(z^{2k-1})<x^{2k},y^{2k}><x,y>^{k-1} of R. In this thesis, we prove that (q_n:m) and q_n+I_n are indeed equal. In the process of proving this equality, we give a Groebner basis for the ideals q_n+I_n and find the corner elements of the monomial ideal <LM(q_n)>.

For every even integer n=2k, let q_n be the ideal <x^{2n},y^{2n},(xy+z^2)^n,z^n> in the polynomial ring R=Q[x,y,z]. In her master's thesis [Y], Yao gives a Groebner basis G for q_n and proves that q_n+I_n contain in (q_n:m), where m is the maximal ideal <x,y,z> of R and I_n is the monomial ideal (x^k)(y^k)(z^{2k-1})<x^{2k},y^{2k}><x,y>^{k-1} of R. In this thesis, we prove that (q_n:m) and q_n+I_n are indeed equal. In the process of proving this equality, we give a Groebner basis for the ideals q_n+I_n and find the corner elements of the monomial ideal <LM(q_n)>.

1. David Cox, John Little, and Donal O'shea, Ideals, Varieties, and Algorithms, Springer-Verge, Heidelberg, Berlinm, 1992.

2. L.-M. Li, Some Algorithms of Corner-Elements in Monomial Ideals, master thesis, National Taiwan Normal University, 2007.

3.Y.-T. Yao, An Application of Groebner Bases : Computing the index of reducibility, master thesis, National Taiwan Normal University, 2009.