Associated Primes And Heights In Ideals Generated By R Elements
Introduction
In the realm of commutative algebra, understanding the structure and properties of ideals within Noetherian rings is paramount. This article delves into a crucial aspect of ideal theory, specifically addressing the question of whether every associated prime of an ideal generated by r elements has a height at most r. This exploration is rooted in the fundamental concepts of Krull dimension, minimal primes, and the powerful Generalized Krull Height Theorem. We aim to provide a comprehensive discussion, clarifying the theorem's implications and offering insights into the behavior of associated primes in the context of ideals generated by a finite number of elements.
The Generalized Krull Height Theorem and Its Implications
The cornerstone of our discussion is the Generalized Krull Height Theorem, a cornerstone result in commutative algebra. This theorem provides a crucial bound on the height of minimal primes ideals lying over an ideal. Let's consider a commutative Noetherian ring, denoted as R, and an ideal I within R that is generated by r elements. The Generalized Krull Height Theorem asserts that for any minimal prime ideal P containing I, the height of P, denoted as ht(P), is at most r. This theorem is a powerful tool for understanding the dimensionality of algebraic varieties and the complexity of solving systems of polynomial equations. To fully appreciate its significance, let's unpack the key concepts involved.
First, we need to define the height of a prime ideal. The height of a prime ideal P in a ring R is the supremum of the lengths of all chains of prime ideals descending from P. In simpler terms, it measures how many prime ideals we can nest strictly between P and a minimal prime ideal. Next, a minimal prime ideal over an ideal I is a prime ideal that contains I and does not contain any smaller prime ideal also containing I. These minimal primes are crucial because they represent the irreducible components of the algebraic set defined by I. The Generalized Krull Height Theorem essentially bounds the complexity of these components based on the number of generators of the ideal I. For instance, if an ideal is generated by a single element (r = 1), any minimal prime over it can have a height of at most 1. This implies a relatively simple geometric structure. However, the theorem does not directly address the height of all associated primes, which is the central question we are investigating. The associated primes, unlike minimal primes, may not directly contain the ideal I, and their relationship to the generators of I is more intricate. Understanding this distinction is essential for navigating the nuances of the question at hand.
Associated Primes: A Deeper Dive
To fully address the central question, we must delve into the concept of associated primes. While the Generalized Krull Height Theorem provides a bound on the height of minimal primes over an ideal, the behavior of associated primes is more nuanced. Let R be a Noetherian ring, and let I be an ideal of R. An associated prime of I is a prime ideal P of R such that P is the annihilator of some element in R/I. In other words, there exists an element x in R that is not in I, but when multiplied by all elements of P, the result falls within I. The set of all associated primes of I is denoted by Ass(R/I). Associated primes play a crucial role in understanding the structure of the ideal I and the module R/I. They provide information about the primary decomposition of I, which is a way of expressing I as an intersection of primary ideals. The minimal primes over I are always associated primes of I, but the converse is not necessarily true. There can be embedded primes, which are associated primes that are not minimal. These embedded primes represent hidden or less obvious aspects of the ideal's structure. The question of whether all associated primes of an ideal generated by r elements have height at most r is a delicate one. The Generalized Krull Height Theorem only guarantees this bound for minimal primes, not for all associated primes. Therefore, a more sophisticated approach is needed to explore this question fully. This involves examining the relationship between the generators of the ideal and the annihilators that define the associated primes. It also requires a careful consideration of the ring's properties, particularly its Noetherian nature, which ensures the existence of primary decompositions and the finiteness of the set of associated primes.
Addressing the Central Question: Height of Associated Primes
Now, let's directly address the question: Does every associated prime of an ideal generated by r elements have height at most r? While the Generalized Krull Height Theorem guarantees this bound for minimal primes over the ideal, it does not extend directly to all associated primes. In general, the answer is no. There exist examples where associated primes of an ideal generated by r elements have a height greater than r. This highlights a critical distinction between minimal primes and associated primes. Minimal primes represent the essential components of the ideal, while associated primes capture finer details of its structure, including embedded components that can exhibit higher dimensionality. To understand why this discrepancy arises, we need to consider the concept of embedded primes. An embedded prime is an associated prime that is not a minimal prime. These embedded primes correspond to components of the ideal that are
