在C-Y. Jean Chan和Jung-Chen Liu一篇發表過的論文中，提出一個充分條件，我們可以得知在二維局部化多項式環k[x,y](x,y)中，符合上述充分條件的理想，就會是特定單項式理想的一個reduction，而這個特定的單項式理想即是將原本理想的生成元之中的加號拿掉後的單項式所生成的。在本篇學位論文中，我們用另一個方法來證明這個充分條件，並且用我們的方法還可以進一步將充分條件推廣到三維的局部化多項式環k[x,y,z](x,y,z).

We consider monomial ideals in the two-dimensional localized polynomial ring k[x,y](x,y) where k is an infinite field. In C-Y. Jean Chan and Jung-Chen Liu's paper, they determine a sufficient condition under which an ideal containing x^a y^b + x^c y^d is a reduction of an ideal containing x^a y^b and x^c y^d. In this thesis, we use another approach to prove the above result. Furthermore, we extend the sufficient condition to the three-dimensional localized polynomial ring k[x,y,z](x,y,z) where k is an infinite field.

[AM] M. F. Atyiah, and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.
[CLO] D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, Springer-Verge, Heidelberg, Berlin, 1992.
[SH] I. Swanson and C. Huneke, Integral Closure of Ideals, Rings, and Modules, Cambridge University Press, Cambridge, 2006.
[CL] C-Y. J. Chan, and J.-C. Liu, A Note on Reductions of Monomial Ideals in k[x, y](x,y), Contemporary Mathematics of AMS, PASI proceedings, 555(2011), 13–34.
[Q1] V. C. Qui˜nonez, Minimal reductions of monomial ideals, Research Reports in Mathematics, Number 10 (2004), Department of Mathematics, Stockholm University, available at http://www.math.su.se/reports/2004/10/.
[Q2] V. C. Qui˜nonez, Integrally Closed Monomial Ideals and Powers of Ideals, Research Reports in Mathematics, Number 7 (2002), Department of Mathematics, Stockholm University, available at http://www.math.su.se/reports/2002/7/.