Assorted Rings and Ideals

Polynomial Rings as Graded rings, and some ideals.

  1. The ring k[x,y]k[x, y] is a graded ring, with grading given by total degree. The homogeneous elements are homogeneous polynomials (polynomials where the total degree of each term is the same).
  2. The ideal (x,y)(x, y) is a homogeneous, prime, and maximal ideal. It is also the irrelevent ideal.
  3. The ideal (x,y2)(x, y^2) is homogeneous.
  4. The ideal (x+y2)(x + y^2) is not homogeneous. It cannot be generated by homogeneous elements. The same is true of (x−a,y−b)(x-a, y-b) for any a,ba, b.