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Krull's principal ideal theorem

Krull's principal ideal theorem is a fundamental result in commutative algebra stating that if R is a Noetherian ring and x \in R, then every prime ideal minimal over the principal ideal (x) has height at most 1.^[1] Named after the German mathematician Wolfgang Krull (1899–1971), the theorem was established in his 1928 paper Primidealketten in allgemeinen Ringbereichen, published in the Sitzungsberichte der Heidelberger Akademie der Wissenschaften.^[2] This work advanced the understanding of ideal chains and dimensions in general rings, building on earlier developments in Noetherian rings by Emmy Noether. The theorem serves as a cornerstone for dimension theory, providing a bound on the codimension of hypersurfaces defined by single elements and enabling proofs of key properties such as the existence of regular sequences and the structure of unique factorization domains in Noetherian settings.^[1] It generalizes to Krull's height theorem, which states that in a Noetherian ring, the height of a minimal prime over an ideal generated by n elements is at most n.^[3] Proofs typically rely on localization, Nakayama's lemma, and properties of Artinian rings, highlighting the theorem's deep connections to primary decomposition and associated primes.^[3]

Background Concepts

Noetherian Rings and Modules

A Noetherian ring is a fundamental concept in commutative algebra, providing a framework for ideals that ensures finite generation and chain stabilization. Specifically, a commutative ring R is Noetherian if every ascending chain of ideals I_1 \subseteq I_2 \subseteq \cdots in R stabilizes, meaning there exists some n such that I_k = I_n for all k \geq n.^[4] This ascending chain condition (ACC) on ideals is equivalent to the property that every ideal in R is finitely generated as an R-module.^[4] Extending this to modules, a module M over a ring R is Noetherian if every ascending chain of submodules N_1 \subseteq N_2 \subseteq \cdots in M stabilizes after finitely many steps.^[5] Equivalently, every submodule of M is finitely generated.^[5] For rings, being Noetherian as a module over itself aligns with the ring definition. These properties underpin much of ideal theory, including behaviors of special ideals like primes that arise in theorems on ring dimension. Key properties of Noetherian rings include the Hilbert basis theorem, which states that if R is Noetherian, then the polynomial ring R (and more generally R[x_1, \dots, x_n]) is also Noetherian.^[6] This theorem, originally proved by David Hilbert in 1890, ensures that polynomial extensions preserve the Noetherian condition, facilitating the study of algebraic varieties.^[6] Classic examples of Noetherian rings include the ring of integers \mathbb{Z}, where every ideal is principal and thus finitely generated, and polynomial rings k[x_1, \dots, x_n] over a field k, which are Noetherian by the Hilbert basis theorem applied iteratively.^[4]^[6] In contrast to Noetherian rings, which satisfy the ACC, Artinian rings satisfy the descending chain condition (DCC) on ideals, where every descending chain I_1 \supseteq I_2 \supseteq \cdots stabilizes.^[7] While Noetherian rings emphasize finite ascending structures, Artinian rings focus on finite descending ones, and commutative Artinian rings are precisely those that are Noetherian and of finite length as modules over themselves.^[7]

Height of Prime Ideals

In commutative algebra, the height of a prime ideal \mathfrak{p} in a commutative ring R with identity, denoted \mathrm{ht}(\mathfrak{p}), is the supremum of the lengths of all strictly ascending chains of prime ideals in R that terminate at \mathfrak{p}. Formally,

\mathrm{ht}(\mathfrak{p}) = \sup \bigl\{ n \in \mathbb{N}_0 \;\big|\; \exists\; \mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n = \mathfrak{p}, \text{ where each } \mathfrak{p}_i \text{ is prime in } R \bigr\},

where the length n of a chain counts the number of strict inclusions, and \mathbb{N}_0 includes 0. The height is infinite if the supremum is unbounded (i.e., there exist such chains of arbitrarily large finite length); otherwise, it is finite.^[8] Such chains typically descend to minimal prime ideals of R, or to the zero ideal when it is prime, as in integral domains.^[9] The height of a prime ideal \mathfrak{p} relates directly to the codimension of the corresponding irreducible closed subset V(\mathfrak{p}) in the spectrum \mathrm{Spec}(R), and the Krull dimension of the ring R, denoted \dim R, is defined as the supremum of the heights of all maximal ideals in R:

\dim R = \sup \bigl\{ \mathrm{ht}(\mathfrak{m}) \;\big|\; \mathfrak{m} \text{ maximal ideal of } R \bigr\}.

This dimension captures the maximal "complexity" of R in terms of prime ideal chains, with \dim R = \mathrm{ht}(\mathfrak{m}) for any maximal ideal \mathfrak{m} in many standard cases, such as polynomial rings.^[10] In Noetherian rings, where the ascending chain condition holds for ideals (and thus for prime ideals), every prime ideal has finite height, ensuring that all such chains are bounded.^[11] For instance, in an integral domain, the zero ideal (0) (which is prime) has height 0, as no proper chain ascends from it. In the polynomial ring k[x, y] over a field k, maximal ideals like (x - a, y - b) for a, b \in k have height 2, realized by chains such as (0) \subsetneq (x - a) \subsetneq (x - a, y - b).^[12] By the definition of height, every chain of prime ideals in R ascending to \mathfrak{p} has length at most \mathrm{ht}(\mathfrak{p}), with equality possible in Noetherian settings where the supremum is attained.^[13]

Statement of the Theorems

The Principal Ideal Theorem

Krull's principal ideal theorem states that if R is a Noetherian ring and (a) is a proper principal ideal generated by a non-unit element a \in R, then every prime ideal minimal over (a) has height at most one.^[14] This bound on the height reflects the geometric intuition that the zero set of a single non-constant function in an affine variety has codimension at most one.^[15] Equivalently, the ideal (a) is contained in some prime ideal of height zero or one, and no minimal prime ideal over (a) has height greater than one.^[3] The minimal primes over (a) form a subset of the associated primes of the module R/(a), denoted \mathrm{Ass}(R/(a)), and the theorem ensures that these minimal elements in the prime spectrum have dimension bounded above by one relative to the whole ring. In the special case where R is an integral domain and a is a nonzero non-unit, the zero ideal does not contain (a), so every minimal prime over (a) has height exactly one, implying that the height of (a) is one.^[14] For example, in the polynomial ring R = k[x, y] over a field k, the principal ideal (x) has unique minimal prime (x), which has height one as the chain (0) \subset (x) achieves length one.^[3]

The Height Theorem

The height theorem provides a generalization of Krull's principal ideal theorem to ideals generated by an arbitrary finite number of elements. Specifically, let R be a Noetherian ring and let I = (f_1, \dots, f_n) be an ideal of R generated by n elements. Then every prime ideal \mathfrak{p} minimal over I satisfies \mathrm{ht}(\mathfrak{p}) \leq n.^[8] The converse direction also holds: in a Noetherian ring R, every prime ideal of height n is minimal over an ideal generated by n elements.^[16] Krull's principal ideal theorem arises as the special case of the height theorem when n = 1.^[8] The height theorem extends naturally to Noetherian modules. Let R be a Noetherian ring and let M be a finitely generated R-module. The annihilator \mathrm{Ann}_R(M) is a finitely generated ideal of R, and every prime ideal minimal over \mathrm{Ann}_R(M) (equivalently, every minimal element of the support of M) has height at most the minimal number of generators of \mathrm{Ann}_R(M).^[8] For an illustrative example, consider the polynomial ring R = k[x, y, z] over a field k, which has Krull dimension 3. The ideal I = (x, y) is generated by 2 elements and is prime with height 2; since I is minimal over itself, this achieves the bound given by the height theorem.^[8]

Proofs

Proof of the Principal Ideal Theorem

Let R be a Noetherian ring and (a) a proper principal ideal generated by a \in R. Let \mathfrak{p} be a minimal prime ideal containing (a). The goal is to prove that the height of \mathfrak{p}, denoted \mathrm{ht}(\mathfrak{p}), satisfies \mathrm{ht}(\mathfrak{p}) \leq 1.^[3] Without loss of generality, localize at \mathfrak{p}, so assume R is a Noetherian local ring with maximal ideal \mathfrak{m} = \mathfrak{p} R_{\mathfrak{p}} and a \in \mathfrak{m}. In this setting, \mathfrak{m} is the unique minimal prime ideal over the extended ideal (a) R, since any prime containing (a) R corresponds to a prime over (a) in the original ring, and minimality of \mathfrak{p} ensures no smaller such prime exists after localization.^[17] To show \mathrm{ht}(\mathfrak{m}) \leq 1, assume for contradiction that \mathrm{ht}(\mathfrak{m}) > 1. Then there exists a prime ideal \mathfrak{q} such that (0) \subsetneq \mathfrak{q} \subsetneq \mathfrak{m}. Since \mathfrak{m} is minimal over (a) R, it follows that (a) R \not\subseteq \mathfrak{q}, so a \notin \mathfrak{q}. Consequently, a is a unit in the localization R_{\mathfrak{q}}, the Noetherian local ring with maximal ideal \mathfrak{n} = \mathfrak{q} R_{\mathfrak{q}}.^[18] Consider the symbolic powers of \mathfrak{q}, defined as \mathfrak{q}^{(n)} = \mathfrak{q}^n R_{\mathfrak{q}} \cap R for n \geq 1. These form a descending chain \mathfrak{q}^{(1)} \supseteq \mathfrak{q}^{(2)} \supseteq \cdots of ideals in R, which stabilizes by the ascending chain condition in the Noetherian ring R; thus, there exists N such that \mathfrak{q}^{(n)} = \mathfrak{q}^{(N)} for all n \geq N. Each \mathfrak{q}^{(n)} is \mathfrak{q}-primary. Moreover, in the localization, \mathfrak{q}^{(n)} R_{\mathfrak{q}} = \mathfrak{n}^n.^[3] Now examine the chain (a) R + \mathfrak{q}^{(n)} \supseteq (a) R + \mathfrak{q}^{(n+1)} \supseteq \cdots in R, which also stabilizes, say at n \geq N, so (a) R + \mathfrak{q}^{(n)} = (a) R + \mathfrak{q}^{(n+1)}. This implies \mathfrak{q}^{(n)} \subseteq (a) R + \mathfrak{q}^{(n+1)}. From the stabilization, \mathfrak{q}^{(n)} \subseteq (a) + \mathfrak{q}^{(n+1)}. For x \in \mathfrak{q}^{(n)}, write x = y + a z with y \in \mathfrak{q}^{(n+1)}, z \in R. Then a z = x - y \in \mathfrak{q}^{(n)}. Since \mathfrak{q}^{(n)} is \mathfrak{q}-primary and a \notin \mathfrak{q}, it follows that z \in \mathfrak{q}^{(n)}. Thus, \mathfrak{q}^{(n)} = (a) \mathfrak{q}^{(n)} + \mathfrak{q}^{(n+1)}. Localizing at \mathfrak{q}, \mathfrak{n}^n = a \mathfrak{n}^n + \mathfrak{n}^{n+1}. Now, the module M = \mathfrak{n}^n / \mathfrak{n}^{n+1} satisfies M = a M. Since a \in \mathfrak{m} (the maximal ideal of R), by Nakayama's lemma, M = 0, so \mathfrak{n}^n = \mathfrak{n}^{n+1}.^[18]^[19] Applying Nakayama's lemma to the finitely generated R_{\mathfrak{q}}-module \mathfrak{n}^n, which satisfies \mathfrak{n} \cdot \mathfrak{n}^n = \mathfrak{n}^{n+1} = \mathfrak{n}^n, and noting that \mathfrak{n} is the maximal ideal, it follows that \mathfrak{n}^n = 0. Thus, \mathfrak{q}^n R_{\mathfrak{q}} = 0. Hence, R_{\mathfrak{q}} has nilpotent maximal ideal and is an Artinian local ring, so \dim R_{\mathfrak{q}} = 0. But since (0) \subsetneq \mathfrak{q}, \dim R_{\mathfrak{q}} = \mathrm{ht}(\mathfrak{q}) \geq 1, a contradiction.^[19] This contradiction implies no such prime \mathfrak{q} exists, so the only prime ideals in R are (0) and \mathfrak{m}, hence \mathrm{dim} R = 1 and \mathrm{ht}(\mathfrak{m}) = 1. In the general case, if a is a zero-divisor, the height may be 0, but in all cases \mathrm{ht}(\mathfrak{p}) \leq 1. The chain length argument confirms that longer prime chains containing (a) cannot exist beyond length 1.^[17]

Proof of the Height Theorem

The proof of the height theorem consists of two parts: establishing that in a Noetherian ring A, if \mathfrak{a} = (f_1, \dots, f_n) is an ideal generated by n elements, then every minimal prime ideal \mathfrak{p} over \mathfrak{a} satisfies \mathrm{ht}(\mathfrak{p}) \leq n; and the converse, that every prime ideal of height n is minimal over some ideal generated by n elements.^[20] The direct part is proved by induction on n, the number of generators of \mathfrak{a}. For the base case n = 1, the result follows from Krull's principal ideal theorem, which states that every minimal prime over a principal ideal has height at most 1.^[20] Assume the statement holds for ideals generated by n-1 elements, where n > 1. Let \mathfrak{a} = (f_1, \dots, f_n) and \mathfrak{p} a minimal prime over \mathfrak{a}. Let \mathfrak{b} = (f_1, \dots, f_{n-1}). Since \mathfrak{p} \supset \mathfrak{b} and \mathfrak{p} is minimal over \mathfrak{b} + (f_n), there exists a prime ideal \mathfrak{q} \subset \mathfrak{p} that is minimal over \mathfrak{b}. By the induction hypothesis, \mathrm{ht}(\mathfrak{q}) \leq n-1. Now consider the quotient ring A/\mathfrak{q}; the image of \mathfrak{a} in this quotient is the principal ideal generated by the image of f_n, and the image of \mathfrak{p} is a minimal prime over this principal ideal. By the principal ideal theorem applied in A/\mathfrak{q}, the height of \mathfrak{p}/\mathfrak{q} is at most 1. Thus, \mathrm{ht}(\mathfrak{p}) = \mathrm{ht}(\mathfrak{q}) + \mathrm{ht}(\mathfrak{p}/\mathfrak{q}) \leq (n-1) + 1 = n.^[20] For the converse, suppose \mathfrak{p} is a prime ideal of height n in the Noetherian ring A. Consider a strictly decreasing chain of prime ideals \mathfrak{p} = \mathfrak{p}_n \supset \mathfrak{p}_{n-1} \supset \cdots \supset \mathfrak{p}_0 = (0) of length n. The proof proceeds by constructing elements x_1, \dots, x_n \in A such that \mathfrak{p}_i is minimal over (x_1, \dots, x_i) for each i = 1, \dots, n. Start with i=1: by the prime avoidance lemma, there exists x_1 \in \mathfrak{p}_1 not contained in any minimal prime of A other than \mathfrak{p}_1, so \mathfrak{p}_1 is minimal over (x_1). Inductively, assume x_1, \dots, x_i have been chosen so that \mathfrak{p}_i is minimal over (x_1, \dots, x_i). Then, by the prime avoidance lemma applied to the minimal primes over (x_1, \dots, x_i) in \mathfrak{p}_{i+1}, there exists x_{i+1} \in \mathfrak{p}_{i+1} \setminus \bigcup (\text{minimal primes over } (x_1, \dots, x_i)), ensuring \mathfrak{p}_{i+1} is minimal over (x_1, \dots, x_{i+1}) and the heights accumulate to i+1. Thus, \mathfrak{p} = \mathfrak{p}_n is minimal over the ideal generated by n elements. This construction relies on Noetherian induction over the primes in the chain and the finite number of minimal primes over each partial ideal.^[16] The theorem extends briefly to Noetherian modules: if M is a finitely generated module over a Noetherian ring A with minimal number of generators n, then every minimal prime over the annihilator ideal \mathrm{Ann}(M) has height at most n. The proof mirrors the ideal case by considering a free resolution or the primary decomposition of M, where the associated primes of M inherit the height bound via the Nakayama lemma applied to the minimal generators.^[16]

Applications and Generalizations

In Dimension Theory

Krull's height theorem provides a fundamental bound in dimension theory by asserting that in a Noetherian ring R, if I is an ideal generated by n elements, then every prime ideal minimal over I has height at most n. This implies that the codimension of such minimal primes is at most n, yielding a lower bound on the Krull dimension of the quotient ring R/I: specifically, \dim(R/I) \geq \dim R - n in domains or catenary rings where height equals codimension. This bound links to the global dimension of rings, particularly in regular local rings where the embedding dimension equals the Krull dimension, ensuring homological dimensions align with algebraic ones.^[12]^[20] In polynomial rings, such as k[x_1, \dots, x_d] over a field k, the theorem establishes that any ideal generated by n elements has height (codimension) at most n, so the dimension of the quotient is at least d - n. This facilitates dimension computations via the Hilbert-Samuel polynomial, which for a finitely generated graded module over such a ring has degree equal to \dim(R/I) - 1, confirming the theorem's bound on possible dimensions through asymptotic growth rates of Hilbert functions.^[12]^[21] Key consequences include the finiteness of Krull dimension in local Noetherian rings: since the maximal ideal \mathfrak{m} is finitely generated with minimal number of generators \mu(\mathfrak{m}), the height \mathrm{ht}(\mathfrak{m}) = \dim R \leq \mu(\mathfrak{m}) < \infty. Additionally, if an ideal I generated by n elements has height exactly n, then its grade equals n, implying that I contains (and is generated by) a regular sequence of length n.^[20]^[22] For example, in the coordinate ring R = k[x_1, \dots, x_d]/J of an affine variety over a field k, the Krull dimension of R equals the dimension of the variety, and maximal ideals in R have height equal to \dim R, corresponding to points on the variety with codimension equal to the ambient dimension.^[12]

Geometric Interpretations

In algebraic geometry, Krull's principal ideal theorem bridges commutative algebra and the dimension theory of varieties by bounding the codimension of loci defined by principal ideals. For an affine variety X = \Spec R, where R is a finitely generated domain over a field k, the principal ideal (f) generated by a nonzero element f \in R defines the hypersurface V(f) \subset X. The theorem states that every minimal prime ideal over (f) has height at most 1, implying that the irreducible components of V(f) have codimension at most 1 in X. If X is irreducible of dimension d, these components thus have dimension at least d-1; in low-dimensional cases, such as plane curves (d=2), they manifest as curves or points, while in higher dimensions, they form pure codimension-1 hypersurface sections.^[23]^[24] The theorem extends naturally to ideals generated by multiple elements via its corollary, known as Krull's height theorem. For an ideal I = (f_1, \dots, f_n) in the coordinate ring R of an affine variety X \subset \mathbb{A}^m_k of dimension d, every minimal prime over I has height at most n, so the irreducible components of the subvariety V(I) \subset X have codimension at most n and dimension at least d - n. This ensures that intersecting X with n hypersurfaces cannot reduce the dimension by more than n; equality holds when V(I) is a complete intersection, providing the expected dimension for such loci.^[24] In the scheme-theoretic framework, which generalizes classical varieties, the theorem interprets the support of modules over Noetherian rings geometrically. For R Noetherian and I generated by n elements, the spectrum \Spec(R/I) has irreducible components corresponding to minimal primes over I, each of codimension at most n in \Spec R, so the dimension of \Spec(R/I) is at least \dim R - n. This bounds the dimension drop for the closed subscheme defined by I, essential for studying supports of coherent sheaves on schemes.^[23] A illustrative example occurs in projective space \mathbb{P}^m_k, where a hypersurface H = V(f) defined by a single nonzero homogeneous polynomial f in the homogeneous coordinate ring has dimension m-1, reducing the ambient dimension by exactly 1 in line with the theorem. Generalizing, a projective subscheme defined by an ideal generated by n homogeneous elements has codimension at most n, mirroring the affine case but accounting for the projective structure; this underpins intersection theory, such as Bézout's theorem, where the theorem guarantees the proper dimension for counting intersection points of curves or higher varieties.^[25]^[26]

References

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### Extracted Statement of Krull's Principal Ideal Theorem
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Krull's principal ideal theorem in non-Noetherian settings
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1 Krull's Principal Ideal Theorem. Lemma 1.1. Let R be a Noetherian ring and P a prime ideal. For n ∈ N, let P(n) = PnRP ∩ R. Then. P(n)RP = (PRP )n. Proof.Missing: statement | Show results with:statement
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Definition 1.1. A commutative ring R is called Noetherian if each ideal in R is finitely generated.
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Definition 1.1 Let R be a commutative ring and M an R-module. We say that M is noetherian if every submodule of M is finitely generated. There is a convenient ...
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Every finitely generated algebra over a noethe- rian ring is noetherian. PROOF. Let A be a noetherian ring, and let B be a finitely generated A-algebra. We.<|control11|><|separator|>
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Section 10.53 (00J4): Artinian rings—The Stacks project
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Definition 9.39. The height of a prime ideal p of a ring R is the supremum of the length n of chains p0 ( p1 ( ··· ( pn = p of prime ideals. Remark 9.40. We ...
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The height of a prime ideal P in a ring R is the supremum of the lengths of (saturated) chains of primes in R that end in P: height(P) := sup{h | ∃ a ...
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Theorem (Krull's principal ideal theorem). Let R be a Noetherian ring, x ∈ R, and. P a minimal prime of xR. Then the height of P ≤ 1.<|control11|><|separator|>
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Theorem 2.1 (Krull's Principal Ideal Theorem). Let R be a Noetherian ring, and pick x ∈ R. Let p be minimal amongst all prime ideals in R containing x.
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REMARKS ON REGULAR SEQUENCES
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On the other hand, by the Krull Principal Ideal Theorem, tr degkk(Y ) = tr degkk(X) − 1. By induction, then, for any closed, irreducible Z ⊂ X, there is a ...
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Sep 20, 2017 · By Krull's Principal Ideal Theorem, it follows that each such minimal prime ideal pi has height one in A(Y ). As A(Y ) is finitely generated ...