Primary Ideal

Definition

A primary ideal is an ideal in a ring such that if the product of two elements belongs to the ideal, at least one of those elements must be in the ideal or the product of the second element with some power of a prime element in the ring also belongs to the ideal. This concept is essential for understanding how ideals can be decomposed into simpler components, which leads to primary decomposition and the identification of associated primes in a given ring.

5 Must Know Facts For Your Next Test

1. In a Noetherian ring, every ideal can be expressed as a finite intersection of primary ideals.
2. The radical of a primary ideal is a prime ideal, which connects primary ideals with prime ideals in a significant way.
3. If an ideal is primary, then any zero divisor in its quotient ring corresponds to a prime element in its decomposition.
4. Primary ideals are critical for understanding the structure of rings and modules through their associated primes.
5. In geometric terms, primary ideals can be linked to irreducible varieties, where each primary component corresponds to a closed point or subvariety.

Review Questions

• How does the concept of primary ideals relate to the notion of decomposing an ideal into simpler components?
  • Primary ideals allow us to decompose more complex ideals into simpler building blocks through primary decomposition. When we express an ideal as an intersection of primary ideals, we can analyze its structure more easily. This decomposition not only simplifies our understanding but also links the original ideal with its associated primes, providing insights into its properties and relationships within the ring.
• Discuss the relationship between primary ideals and their radicals in terms of their connection to prime ideals.
  • The radical of a primary ideal is directly related to prime ideals in that it forms a prime ideal. This means that when we take the radical of a primary ideal, we can discover essential properties about the original ideal itself. Understanding this relationship helps us better analyze the structure and behavior of both primary and prime ideals within a ring.
• Evaluate how primary ideals contribute to our understanding of algebraic varieties and their irreducibility.
  • Primary ideals play a crucial role in understanding algebraic varieties by linking algebraic concepts to geometric intuition. Each primary component corresponds to an irreducible variety, allowing us to connect algebraic properties of these ideals with the geometric structures they represent. This connection enriches our understanding of how algebraic geometry operates, particularly regarding the classifications and characteristics of varieties.

Related terms

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

Primary Decomposition: The process of expressing an ideal as an intersection of primary ideals, which helps analyze its structure and associated primes.

Associated Prime: A prime ideal that corresponds to a primary ideal in a decomposition, providing insight into the structure of the ring and its modules.