Associated prime

Definition

An associated prime is a prime ideal that is linked to a particular element or ideal through the concept of primary decomposition. This term is crucial for understanding how ideals can be decomposed into primary components, with associated primes representing the 'top' or 'most important' prime factors in this structure. They provide insight into the structure of the ring and its modules, revealing key properties such as depth and height.

5 Must Know Facts For Your Next Test

1. Associated primes play a critical role in understanding the localization and support of modules, revealing where their non-zero elements reside.
2. Every primary ideal has an associated prime, and conversely, associated primes are linked to primary components in primary decomposition.
3. If a module has an associated prime, it can be decomposed into simpler components using these primes as a foundation for understanding its structure.
4. Associated primes can give insights into the geometric properties of varieties, particularly in algebraic geometry where they relate to irreducible components.
5. The set of associated primes of a module can be computed using the radical of its annihilator, connecting algebraic properties with geometric interpretations.

Review Questions

• How do associated primes relate to primary ideals and their decomposition?
  • Associated primes are integral to understanding primary ideals since every primary ideal corresponds to one or more associated primes. When you perform primary decomposition on an ideal, you break it down into primary components, and each component will have its own associated prime. This relationship helps in analyzing the structure of ideals and their behavior within rings, making it easier to study more complex algebraic constructs.
• Discuss the significance of associated primes in determining the depth and height of prime ideals.
  • Associated primes significantly influence both depth and height in ring theory. The height of a prime ideal reflects how many levels of prime ideals are below it, while associated primes can impact this by determining which prime ideals contribute to a given structure. The depth relates to how many elements can form a regular sequence before hitting an associated prime, thus shaping our understanding of module theory and its implications on algebraic structures.
• Evaluate how associated primes can affect the geometric interpretation of algebraic varieties and their irreducible components.
  • Associated primes provide essential insights into the geometric properties of algebraic varieties, particularly by highlighting irreducible components. Each associated prime corresponds to a unique component in the variety's structure, meaning that they help us identify where varieties may intersect or overlap. By evaluating these associations, mathematicians can better understand complex interactions within geometric spaces, allowing for deeper analyses in both algebraic geometry and commutative algebra.

Related terms

Primary ideal: An ideal is primary if whenever the product of two elements is in the ideal, at least one of those elements must be in the ideal or is nilpotent.

Height of a prime ideal: The height of a prime ideal is the length of the longest chain of prime ideals contained within it, indicating how 'deep' the prime ideal is within the overall structure of the ring.

Depth of a module: The depth of a module measures the minimum number of elements needed to generate a regular sequence, reflecting the complexity of its structure relative to associated primes.