Primary Ideal

Definition

A primary ideal is an ideal in a ring such that if the product of two elements belongs to the ideal, then at least one of the elements is in the ideal or some power of the other element is in the ideal. This concept connects deeply with the structure of rings and ideals, as it allows for a refined understanding of their decomposition and associated prime ideals.

5 Must Know Facts For Your Next Test

1. Every prime ideal is also a primary ideal, but not every primary ideal is prime.
2. The radical of a primary ideal is a prime ideal, which reflects how primary ideals relate to the underlying prime structure.
3. In a Noetherian ring, any ideal can be expressed as an intersection of primary ideals, which shows how primary ideals aid in understanding the ring's structure.
4. Primary decomposition allows us to represent an ideal as an intersection of primary ideals, each associated with specific prime ideals.
5. The associated primes of a primary ideal are important for understanding its composition and provide valuable information about the variety defined by the ideal.

Review Questions

• How do primary ideals relate to prime ideals within a ring?
  • Primary ideals and prime ideals are closely related concepts in ring theory. A primary ideal is defined such that if a product belongs to it, one of the factors is in the ideal or some power of the other is. This means that every prime ideal fits into the definition of a primary ideal; however, primary ideals may not necessarily be prime themselves. Understanding this relationship helps clarify how different types of ideals interact within rings.
• Discuss how primary decomposition can be applied to an ideal in a Noetherian ring and what implications this has for associated primes.
  • In a Noetherian ring, every ideal can be expressed as an intersection of primary ideals through primary decomposition. This process not only simplifies the structure of the ideal but also reveals its associated primes, which are crucial for understanding the algebraic properties and geometric interpretations tied to that ideal. The ability to decompose into primary components allows for effective analysis and helps in determining properties such as dimension and irreducibility.
• Evaluate the significance of associated primes in relation to primary ideals and how they contribute to our understanding of algebraic varieties.
  • Associated primes provide essential insights into the structure of primary ideals by indicating which prime ideals contribute to the formation of those components. In terms of algebraic varieties, these associated primes help delineate critical features like singularity and irreducibility. By evaluating how primary ideals decompose into their associated primes, we gain deeper insight into the underlying geometric structures represented by these ideals, ultimately enriching our understanding of algebraic geometry.

Related terms

Prime Ideal: An ideal in a ring such that if the product of two elements is in the ideal, then at least one of those elements must also be in the ideal.

Associated Prime:

Primes that are associated with a given ideal, playing a key role in understanding the structure of modules over a ring.

Radical of an Ideal:

The set of elements in a ring that when raised to some power belong to a specific ideal, providing insights into its structure and related prime ideals.