Construction of Primary Ideals and Their Association with Primary Ideal Rings

Introduction

In commutative algebra, primary ideals play a pivotal role in understanding the structure of rings, particularly in the study of ideals that capture the behavior of prime ideals and their extensions. A primary ideal is an ideal $ P $ in a ring $ R $ such that $ P = (a_1, a_2, \ldots, a_n) $ satisfies the condition that if $ x \in R $ and $ x \notin P $, then $ x^n \in P $ for some positive integer $ n $. This property ensures that primary ideals are closely tied to the radical of the ideal, and they generalize the concept of prime ideals by introducing a more nuanced relationship between containment and nilpotency.

The study of primary ideals extends to the concept of primary ideal rings, which are rings in which every prime ideal is also a primary ideal. Such rings exhibit a stronger form of coherence, often leading to the resolution of singularities and the classification of modules. This article explores the construction of primary ideals, their defining properties, and their association with primary ideal rings, emphasizing their significance in modern ring theory.

Definition and Properties of Primary Ideals

A primary ideal in a ring $ R $ is an ideal $ P $ such that for any elements $ a, b \in R $, if $ ab \in P $, then either $ a \in P $ or $ b \in P^{(n)} $ for some positive integer $ n $, where $ P^{(n)} $ denotes the $ n $-th power of the ideal $ P $. This condition ensures that the ideal $ P $ is "closed" under the operation of taking powers of elements that are not in $ P $, thereby capturing the essence of nilpotent elements.

Primary ideals are closely related to prime ideals. In a ring with a unit, a prime ideal $ P $ is also a primary ideal if and only if $ P $ is a maximal ideal. This connection underscores the importance of primary ideals in the study of Noetherian rings, where the prime ideals are precisely the primary ideals. Moreover, in a ring with a unit, the radical of a primary ideal coincides with the ideal itself, a property that simplifies the analysis of ideals in such rings.

Construction of Primary Ideals

The construction of primary ideals often involves the use of specific ring structures, such as polynomial rings, Dedekind domains, and rings of fractions. For example, in a polynomial ring $ R[x] $, primary ideals can be constructed by considering the ideal generated by a polynomial and its derivatives, ensuring that the ideal is primary. Similarly, in a Dedekind domain, primary ideals correspond to prime ideals, and their construction relies on the unique factorization of ideals.

In general, the construction of primary ideals involves defining an ideal $ P $ such that it satisfies the primary condition. This can be achieved through various methods, including the use of radical ideals, the study of associated primes, and the application of the Nullstellensatz. For instance, in a ring $ R $ with a unit, the primary ideals can be generated by elements that are not necessarily prime, yet their powers are contained within the ideal. This flexibility allows for a rich structure in the study of ideals in non-Noetherian rings.

Primary Ideal Rings and Their Properties

A primary ideal ring is a ring in which every prime ideal is also a primary ideal. This property is particularly significant in the study of rings with a unit, where the structure of ideals is more constrained. In such rings, the radical of an ideal coincides with the ideal itself, and the primary condition ensures that the ideal is "closed" under the operation of taking powers of elements not in the ideal.

The construction of primary ideal rings often involves the use of specific ring constructions, such as the ring of integers, the ring of formal power series, and the ring of fractions. For example, in the ring of integers $ \mathbb{Z} $, every prime ideal is also a primary ideal, reflecting the unique factorization properties of integers. Similarly, in the ring of formal power series $ \mathbb{C}[[x]] $, the primary ideals correspond to prime ideals, and their construction relies on the structure of the ring.

In general, the construction of primary ideal rings involves ensuring that the ring satisfies the condition that every prime ideal is primary. This can be achieved through the use of specific ring properties, such as the existence of a unit, the finiteness of the number of prime ideals, and the compatibility of ideals with the ring's structure. Such rings often exhibit a strong form of coherence, allowing for the classification of modules and the resolution of singularities.

Relationship Between Primary Ideals and Primary Ideal Rings

The relationship between primary ideals and primary ideal rings is fundamental to the study of commutative algebra. In a ring with a unit, the primary ideals are precisely the ideals for which the radical of the ideal coincides with the ideal itself. This property ensures that the ideal is "closed" under the operation of taking powers of elements not in the ideal, a key feature of primary ideals.

In a primary ideal ring, every prime ideal is also a primary ideal, which further strengthens the structure of the ring. This property is particularly useful in the study of rings with a unit, where the radical of an ideal coincides with the ideal itself. The construction of such rings often involves the use of specific ring constructions, such as the ring of integers, the ring of formal power series, and the ring of fractions, which ensure that the ring satisfies the necessary conditions for the primary ideal property.