Associated Prime

Definition

An associated prime of a ring is a prime ideal that corresponds to the zero divisors of a module over that ring, indicating where the module fails to be free. These primes reveal important structural information about the module and are closely related to primary decomposition, as they help classify the components of the module into more manageable pieces. Understanding associated primes provides insights into the depth and regularity of modules, especially in the context of Cohen-Macaulay rings.

5 Must Know Facts For Your Next Test

1. Associated primes can be seen as a bridge between algebraic geometry and commutative algebra, linking geometric properties to algebraic structures.
2. For any finitely generated module over a Noetherian ring, the associated primes are precisely the primes that correspond to its primary components in a primary decomposition.
3. The number of associated primes can reveal information about the complexity of a module; more associated primes typically indicate a more complicated structure.
4. In Cohen-Macaulay rings, all associated primes have the same height, which allows for a uniform understanding of their structure and relationships.
5. Associated primes are particularly important in studying singularities, where they can inform us about the types and characteristics of singular points in geometric spaces.

Review Questions

• How do associated primes contribute to understanding the structure of modules in algebra?
  • Associated primes help in breaking down modules into simpler components by revealing where they fail to be free. This structural insight is essential for analyzing properties like depth and regular sequences. By identifying these primes, one can classify the zero divisors in a module, leading to a clearer understanding of its algebraic characteristics and how it behaves under various operations.
• Discuss the role of associated primes in primary decomposition and their implications for the study of modules over Noetherian rings.
  • Associated primes play a crucial role in primary decomposition by providing the link between primary ideals and their corresponding prime ideals. In any finitely generated module over a Noetherian ring, each primary component contributes an associated prime that encapsulates essential information about that component. This relationship allows mathematicians to analyze modules' structure, particularly when dealing with intricate algebraic objects and their properties.
• Evaluate the significance of associated primes in Cohen-Macaulay rings and how they relate to depth and regular sequences.
  • In Cohen-Macaulay rings, associated primes are not just random; they are uniformly related as all have the same height. This consistency greatly simplifies the study of these rings' modules, allowing for more straightforward application of depth and regular sequence concepts. The alignment of associated primes with depth provides powerful tools for classifying modules' behavior and understanding their geometric interpretations, making them fundamental in both algebra and geometry.

Related terms

Primary Decomposition:

A way to express an ideal as an intersection of primary ideals, where each primary ideal corresponds to a unique associated prime.

Depth:

The length of the longest regular sequence contained in an ideal, which gives information about how far the module is from being free.

Cohen-Macaulay Rings: Rings in which every non-zero finitely generated module has depth equal to its Krull dimension, often characterized by having a well-behaved set of associated primes.