Primary Ideal

Definition

A primary ideal is an ideal in a ring that has the property that if a product of two elements belongs to the ideal, then at least one of those elements must belong to the ideal or is nilpotent. This concept is crucial in understanding the structure of ideals and their correspondence to algebraic varieties, particularly in how primary ideals relate to irreducible components of varieties.

5 Must Know Facts For Your Next Test

1. Every primary ideal can be expressed as the intersection of primary components, which helps in the decomposition of ideals.
2. In a primary ideal, if an element 'a' belongs to the ideal and 'b' is any element such that 'ab' is in the ideal, then either 'a' is in the ideal or 'b' is nilpotent.
3. The radical of a primary ideal corresponds to its associated prime ideal, which provides insight into its structure and properties.
4. Primary ideals play a significant role in the study of Noetherian rings and their properties, particularly in terms of dimension theory.
5. The correspondence between primary ideals and algebraic varieties helps illustrate how algebraic properties translate into geometric ones.

Review Questions

• How do primary ideals contribute to our understanding of irreducible components of algebraic varieties?
  • Primary ideals are closely linked to irreducible components since every irreducible component of an algebraic variety corresponds to a primary ideal. Specifically, if you have an algebraic variety represented by an ideal, the primary decomposition reveals how this variety can be broken down into irreducible components. Each primary component reflects a distinct geometric feature of the variety, allowing for a deeper understanding of its structure through the lens of algebra.
• Compare and contrast primary ideals with radical ideals, highlighting their differences in terms of containment conditions.
  • Primary ideals and radical ideals differ primarily in their containment conditions. In a primary ideal, if a product 'ab' belongs to the ideal, at least one of 'a' or 'b' must be either in the ideal or nilpotent. In contrast, for a radical ideal, if any power of an element 'a^n' is in the ideal, then 'a' itself must also be in it. This distinction highlights how primary ideals deal with factors involving nilpotency, whereas radical ideals focus on roots and their membership within the ideal.
• Evaluate the importance of primary ideals in the context of Noetherian rings and dimension theory.
  • Primary ideals are fundamental to understanding Noetherian rings as they allow us to explore their structure through primary decompositions. In dimension theory, primary ideals help determine the dimensions of varieties and their associated schemes by providing information about chains of prime ideals and their heights. This evaluation allows mathematicians to understand not just the algebraic properties but also the geometric implications within Noetherian rings, revealing intricate relationships between algebra and geometry.

Related terms

Nilpotent Element: An element in a ring that becomes zero when raised to some power.

Radical Ideal:

An ideal such that if an element's power is in the ideal, then the element itself is also in the ideal.

Irreducible Variety:

An algebraic variety that cannot be expressed as a union of two proper subvarieties.