(x^2, y)

Definition

The ideal (x^2, y) in the polynomial ring k[x,y] is generated by the polynomials x^2 and y, which means it consists of all linear combinations of these generators with coefficients in the field k. This ideal is a key example when discussing primary ideals, as it has specific algebraic properties that help in understanding how ideals behave within a ring and their role in factorization and localization.

5 Must Know Facts For Your Next Test

1. The ideal (x^2, y) is primary because its zero set corresponds to the point (0, 0), making it a primary component related to the variable y.
2. In the context of primary ideals, (x^2, y) can be decomposed into prime ideals, specifically related to the ideal generated by y and the ideal generated by x.
3. This ideal is not maximal, as it does not correspond to a field but rather to a local ring whose maximal ideal contains (x^2, y).
4. The generators of this ideal lead to certain algebraic properties such as nilpotent elements that arise from powers of x in the context of localization.
5. The structure of (x^2, y) showcases how polynomial ideals can intersect with geometric intuition by representing points on curves in the affine plane.

Review Questions

• How does the ideal (x^2, y) relate to the concept of primary ideals in terms of its generators and their properties?
  • The ideal (x^2, y) is a primary ideal because its generators have specific multiplicative relationships that reflect the characteristics of primary ideals. In particular, if we have an element multiplied by another element belonging to this ideal, then at least one of these elements must belong to the ideal itself or its power must do so. This links back to how we understand the intersection of geometric properties with algebraic ideals, especially focusing on points in space represented by these generators.
• Discuss how the structure of the ideal (x^2, y) helps in understanding radical ideals and their applications in algebraic geometry.
  • The structure of (x^2, y) gives insight into radical ideals because it highlights how certain elements relate through their powers. The ideal's nature indicates that while x^2 is included in this ideal, its root x must also belong to some larger context of an associated prime ideal. This interplay assists in identifying geometric objects represented by these ideals within algebraic geometry, particularly when considering their intersection with points in the affine plane.
• Evaluate the significance of localization with respect to the ideal (x^2, y) and how it affects our understanding of polynomial rings.
  • Localization regarding (x^2, y) allows us to focus on properties specific to this ideal within k[x,y], enhancing our understanding of its behavior under different conditions. By localizing at prime ideals associated with (x^2) or (y), we can simplify complex relationships into more manageable parts. This process reveals details about singularities and behavior near (0,0), which is crucial for both computational and theoretical aspects in algebraic geometry and commutative algebra.

Related terms

Primary Ideal: An ideal I in a ring R is primary if whenever ab ∈ I for some a, b ∈ R, then either a ∈ I or b^n ∈ I for some n > 0.

Radical Ideal: An ideal I is called radical if whenever x^n ∈ I for some n > 0, then x ∈ I. The radical of an ideal can be related to the structure of varieties.

Localization: Localization is a process that allows one to focus on the behavior of a ring at a particular prime ideal, often simplifying the study of its structure and properties.