Associated prime ideals are prime ideals that correspond to the minimal prime ideals in a primary decomposition of an ideal. These ideals play a crucial role in understanding the structure of modules over a ring, particularly in the context of primary decomposition, where an ideal can be expressed as an intersection of primary ideals, each associated with a prime ideal.
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Each associated prime ideal corresponds to a unique primary component in the primary decomposition of an ideal.
If P is an associated prime ideal of an ideal I, then there exists an element in I such that its annihilator is exactly P.
In a Noetherian ring, every ideal has a finite primary decomposition, leading to a finite set of associated prime ideals.
The set of associated prime ideals can provide insights into the singularities of algebraic varieties represented by the ideals.
The number and nature of associated prime ideals can influence the structure of modules over the ring, especially in terms of their decomposition and representation.
Review Questions
How do associated prime ideals relate to primary decomposition, and why are they important in understanding the structure of ideals?
Associated prime ideals are directly linked to primary decomposition because they represent the minimal primes that arise when decomposing an ideal into primary components. Understanding these associated primes allows us to analyze the underlying structure of the ideal itself, as they reveal information about its factors and behavior. This relationship is essential for grasping how different components interact within a ring and can lead to deeper insights into module theory and algebraic geometry.
Discuss how the presence of associated prime ideals can influence the properties of a module over a Noetherian ring.
The presence of associated prime ideals in a module over a Noetherian ring can significantly impact its structure and properties. Specifically, the composition series of the module can be influenced by these primes, as they dictate the factors that appear during decomposition. Furthermore, associated primes help determine the support of a module, which indicates where it 'lives' in terms of the spectrum of the ring. This allows us to understand not only its algebraic structure but also its geometric implications.
Evaluate the importance of associated prime ideals in the context of algebraic geometry and their role in determining singularities.
Associated prime ideals play a critical role in algebraic geometry by providing insights into singularities of varieties. When considering an algebraic variety defined by an ideal, its associated prime ideals indicate points where the variety may exhibit non-smooth behavior or intersections with multiplicity. These primes help identify not just where singular points occur, but also give information on their nature and how they affect the overall geometry of the variety. Consequently, analyzing associated prime ideals is essential for understanding both local and global properties of algebraic varieties.
Related terms
Primary Ideal: An ideal I in a ring R is primary if whenever ab ∈ I for some a, b ∈ R, then either a ∈ I or b^n ∈ I for some n ≥ 1.
Primary Decomposition: The expression of an ideal as an intersection of primary ideals, providing insight into its structure and properties.
Radical of an Ideal: The radical of an ideal I is the set of all elements r in the ring such that r^n ∈ I for some positive integer n. It helps identify the associated primes.