Associated prime

Definition

An associated prime is a prime ideal that is linked to a module through its elements, indicating a deep connection between the algebraic structure of the module and the geometry of the space it represents. Each associated prime can be seen as a reflection of certain properties of the module, particularly in the context of primary decomposition, where they reveal information about the 'hidden' structure of ideals. This term plays a crucial role in understanding how various ideals relate to one another and how they interact with the underlying algebraic geometry.

5 Must Know Facts For Your Next Test

1. Associated primes are crucial in decomposing modules and understanding their structure, especially in Noetherian rings.
2. Each associated prime corresponds to a minimal prime over the annihilator of a module, revealing important information about the module's properties.
3. In the context of primary decomposition, each associated prime arises from primary components, offering insights into the geometric properties of varieties.
4. The number of associated primes can be finite or infinite depending on the structure of the module and the ring.
5. Associated primes help determine the depth and dimension of modules, which are vital in both algebraic and geometric contexts.

Review Questions

• How do associated primes relate to primary decomposition and what insights do they provide about a module?
  • Associated primes are directly linked to primary decomposition as they emerge from the primary components of an ideal. Each associated prime highlights specific structural aspects of a module, such as its depth and support. This connection helps understand how various ideals interact within the algebraic framework and allows mathematicians to reveal hidden geometric properties related to varieties represented by those modules.
• Discuss how associated primes can influence our understanding of a module's dimension and depth.
  • Associated primes play a significant role in determining both the dimension and depth of modules. The dimension can be inferred from the set of associated primes, where their variety relates directly to the geometrical interpretation of dimension. Additionally, depth, which measures how 'far' a module is from being free, is influenced by the number and nature of associated primes. Analyzing these primes gives insight into whether the module can be resolved or how it behaves under extensions.
• Evaluate the implications of associated primes in both algebraic and geometric contexts and how they contribute to our overall understanding of algebraic structures.
  • The implications of associated primes stretch across both algebraic and geometric domains by providing critical information about modules and varieties. In algebra, they facilitate primary decomposition, revealing intrinsic properties of ideals within rings. Geometrically, associated primes correspond to points or subvarieties in schemes, enriching our understanding of spaces defined by these ideals. Ultimately, their study enhances our grasp of how algebraic structures correlate with geometric intuition, bridging these fields effectively.

Related terms

Primary Ideal: An ideal I in a ring R such that if ab ∈ I for some a ∈ R, then either a ∈ I or b^n ∈ I for some n > 0.

Primary Decomposition: The representation of an ideal as an intersection of primary ideals, which provides insight into its structure and properties.

Support: The set of prime ideals associated with a module that indicates where the module 'lives' in the spectrum of the ring.