Krull dimension; Krull's principal ideal theorem

Intuition [Beginner]

Think of a ring as a landscape with special locations called prime ideals. Some sit inside bigger ones, forming nested chains like Russian dolls. The Krull dimension of the ring is the length of the longest chain: how many strict containments the tallest chain has.

For the integers $\mathbb{Z}$, the chain goes from $(0)$ up to $(2)$, the ideal of even numbers. Since $2$ is prime, the ideal $(2)$ is maximal among proper ideals. The longest chain has one link, so $\mathbb{Z}$ has Krull dimension $1$.

Why does this concept exist? It measures how many independent degrees of freedom a ring has. A polynomial ring in $n$ variables has Krull dimension $n$: each new variable adds one nesting level. Krull's theorem says that imposing one equation can reduce the dimension by at most one, mirroring the geometric fact that a single constraint cuts down the free directions by at most one.

Visual [Beginner]

Three concentric regions representing a chain $(0)\subset P_1\subset P_2$ of three nested prime ideals in a ring of Krull dimension $2$. The innermost disc is the zero ideal $(0)$, the middle annulus is $P_1$, and the outer annulus is $P_2$. Arrows mark the strict containments between successive regions.

The nesting depth of the largest chain fixes the dimension of the ring.

Worked example [Beginner]

Consider the ring $k[x,y]$ of polynomials in two variables over a field $k$.

Step 1. The zero ideal $(0)$ is prime (the quotient is $k[x,y]$, a domain). The ideal $(x)$ is prime (the quotient is $k[y]$, a domain). The ideal $(x,y)$ is prime (the quotient is $k$, a field).

Step 2. These ideals form a chain $(0)\subset (x)\subset (x,y)$ with two strict containments, so the chain has length $2$.

Step 3. Pass to the quotient $k[x,y]/(xy)$, imposing the equation $xy=0$. The dimension drops to $1$: the longest chain of primes containing $(xy)$ has length $1$, for instance $(x)\subset (x,y)$. Adding one equation reduced the dimension by exactly one.

What this tells us: each polynomial equation can reduce the dimension of the solution space by at most one, matching the geometric intuition that a hypersurface in $n$-dimensional space has dimension $n-1$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

A prime ideal $\mathfrak{p}$ of a commutative ring $R$ is a proper ideal satisfying: if $ab\in \mathfrak{p}$ then $a\in \mathfrak{p}$ or $b\in \mathfrak{p}$ ^{[Atiyah-Macdonald 1969]}. Equivalently, the quotient ring $R/\mathfrak{p}$ is an integral domain.

A chain of prime ideals of length $n$ is a strictly increasing sequence $$ \mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n $$ where each ${\mathfrak{p}}_i$ is a prime ideal of $R$.

Definition (Krull dimension). The Krull dimension of $R$, denoted $\dim R$, is the supremum of the lengths $n$ of all chains of prime ideals in $R$. If no finite supremum exists, $\dim R=\infty$. The zero ring has $\dim R=-1$ by convention.

Definition (Height). The height of a prime ideal $\mathfrak{p}$, denoted $\mathrm{ht}(\mathfrak{p})$, is the supremum of lengths of chains of prime ideals ${\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_k=\mathfrak{p}$ terminating at $\mathfrak{p}$.

The localisation of $R$ at a prime ideal $\mathfrak{p}$, denoted $R_{\mathfrak{p}}$, is the ring of fractions $a/s$ with $s\notin \mathfrak{p}$, under the usual equivalence $a/s=b/t$ when $u(at-bs)=0$ for some $u\notin \mathfrak{p}$. The prime ideals of $R_{\mathfrak{p}}$ correspond bijectively to the prime ideals of $R$ contained in $\mathfrak{p}$. Consequently $\mathrm{ht}(\mathfrak{p})=\dim R_{\mathfrak{p}}$.

Counterexamples to common slips [Intermediate+]

• Krull dimension is not vector-space dimension. The ring $k[x]/(x^2)$ has Krull dimension $0$ (its only prime ideal is $(x)$) but is $2$-dimensional as a $k$-vector space. The two notions measure different things.

• The supremum may not be attained. Nagata constructed a Noetherian ring of infinite Krull dimension: an infinite family of prime ideals at unbounded heights, each of finite length, with no single infinite chain.

• Adding generators does not always drop dimension. In $k[x,y]$, the ideal $(x^2,xy)$ has two generators but the quotient $k[x,y]/(x^2,xy)$ has dimension $1$, not $0$. The correct statement is Krull's theorem: each generator drops dimension by at most one, and the drop occurs only when the generator is a non-zero-divisor modulo the previous ones.

Key theorem with proof [Intermediate+]

Theorem (Krull's principal ideal theorem). Let $R$ be a Noetherian ring and $a\in R$ a non-unit. If $\mathfrak{p}$ is a prime ideal minimal among all prime ideals containing $(a)$, then $\mathrm{ht}(\mathfrak{p})\le 1$.

Proof. Suppose for contradiction that $\mathrm{ht}(\mathfrak{p})\ge 2$. Choose primes ${\mathfrak{q}}_0⊊\mathfrak{q}⊊\mathfrak{p}$.

Step 1 (Localisation). Replace $R$ by $R_{\mathfrak{p}}$. The resulting ring $S=R_{\mathfrak{p}}$ is a Noetherian local ring with maximal ideal $\mathfrak{n}=\mathfrak{p}S$, which is minimal over $(a/1)$, and the chain ${\mathfrak{q}}_{0}S⊊\mathfrak{q}S⊊\mathfrak{n}$ persists.

Step 2 (Pass to a domain). The quotient $\bar{S}=S/{\mathfrak{q}}_{0}S$ is a Noetherian domain. The image $\bar{a}$ is nonzero (since ${\mathfrak{q}}_{0}S⊊\mathfrak{n}$ and $\mathfrak{n}$ is minimal over $(a/1)$, the element $a/1\notin {\mathfrak{q}}_{0}S$). The image $\bar{\mathfrak{q}}=\mathfrak{q}S/{\mathfrak{q}}_{0}S$ is a nonzero prime strictly contained in $\bar{\mathfrak{n}}=\mathfrak{n}/{\mathfrak{q}}_{0}S$, which is minimal over $(\bar{a})$.

Replacing $\bar{S}$ by its localisation at $\bar{\mathfrak{n}}$, we may assume $(R,\mathfrak{m})$ is a Noetherian local domain, $a\in \mathfrak{m}$ is nonzero, $\mathfrak{m}$ is minimal over $(a)$, and there exists a nonzero prime $\mathfrak{q}⊊\mathfrak{m}$.

Step 3 (Symbolic powers). Define the $n$-th symbolic power of $\mathfrak{q}$ by ${\mathfrak{q}}^{(n)}={\mathfrak{q}}^n{R}_{\mathfrak{q}}\cap R$. Each ${\mathfrak{q}}^{(n)}$ is $\mathfrak{q}$-primary. Since $R_{\mathfrak{q}}$ is a local domain with maximal ideal $\mathfrak{q}{R}_{\mathfrak{q}}$, the Krull intersection theorem gives ${\bigcap }_{n\ge 1}{\mathfrak{q}}^n{R}_{\mathfrak{q}}=0$. Contracting to $R$ yields ${\bigcap }_{n\ge 1}{\mathfrak{q}}^{(n)}=0$. Since $\mathfrak{q}\ne 0$ and ${\mathfrak{q}}^{(1)}=\mathfrak{q}$, the symbolic powers form a strictly descending chain: ${\mathfrak{q}}^{(1)}⊋{\mathfrak{q}}^{(2)}⊋\cdots$.

Since $\mathfrak{m}$ is minimal over $(a)$ and $\mathfrak{q}⊊\mathfrak{m}$, the element $a\notin \mathfrak{q}$. Being a unit in $R_{\mathfrak{q}}$, the element $a$ does not belong to any ${\mathfrak{q}}^{(n)}$.

Step 4 (Stabilisation argument). Since $R$ is Noetherian, the ascending chain of ideals $(a)+{\mathfrak{q}}^{(1)}\supset (a)+{\mathfrak{q}}^{(2)}\supset \cdots$ stabilises at some index $N$: for all $n\ge N$, $$ (a) + \mathfrak{q}^{(N)} = (a) + \mathfrak{q}^{(n)}. $$

Take any $b\in {\mathfrak{q}}^{(N)}$. Since $b\in (a)+{\mathfrak{q}}^{(N+1)}$, write $b=ca+d$ with $c\in R$ and $d\in {\mathfrak{q}}^{(N+1)}\subset {\mathfrak{q}}^{(N)}$. Then $ca=b-d\in {\mathfrak{q}}^{(N)}$. The ideal ${\mathfrak{q}}^{(N)}$ is $\mathfrak{q}$-primary (as the contraction of ${\mathfrak{q}}^N{R}_{\mathfrak{q}}$, which is $\mathfrak{q}{R}_{\mathfrak{q}}$-primary). Since $a\notin \mathfrak{q}=\sqrt{{\mathfrak{q}}^{(N)}}$, the primary property forces $c\in {\mathfrak{q}}^{(N)}$.

Therefore $b=ca+d$ with $c\in {\mathfrak{q}}^{(N)}$ and $d\in {\mathfrak{q}}^{(N+1)}$, which gives ${\mathfrak{q}}^{(N)}=a\cdot {\mathfrak{q}}^{(N)}+{\mathfrak{q}}^{(N+1)}$.

Step 5 (Nakayama contradiction). Consider the finitely generated $R$-module $M={\mathfrak{q}}^{(N)}/{\mathfrak{q}}^{(N+1)}$. The identity ${\mathfrak{q}}^{(N)}=a\cdot {\mathfrak{q}}^{(N)}+{\mathfrak{q}}^{(N+1)}$ translates to $M=aM$. By Nakayama's lemma (applicable because $R$ is local with maximal ideal $\mathfrak{m}$, the module $M$ is finitely generated, and $a\in \mathfrak{m}$), this forces $M=0$, i.e.\ ${\mathfrak{q}}^{(N)}={\mathfrak{q}}^{(N+1)}$.

This contradicts the strict descent ${\mathfrak{q}}^{(N)}⊋{\mathfrak{q}}^{(N+1)}$ established in Step 3. Therefore $\mathrm{ht}(\mathfrak{p})\le 1$. $\square$

Bridge. The central insight behind Krull's principal ideal theorem is that symbolic powers of a prime ideal, combined with Nakayama's lemma in the localised ring, force the height constraint $\mathrm{ht}(\mathfrak{p})\le 1$. This builds toward the dimension formula for polynomial rings 01.02.17, where the iterated Hilbert basis theorem provides the Noetherian hypothesis, and appears again in the generalized principal ideal theorem as the inductive step. The foundational reason is that the Noetherian condition turns the infinitary problem of bounding chain lengths into a finitary one, and the bridge is between the algebraic data of ideal generation and the geometric data of codimension.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Generalized principal ideal theorem). Let $R$ be a Noetherian ring, $a_1,\dots ,a_n\in R$, and $\mathfrak{p}$ a prime minimal over $(a_1,\dots ,a_n)$. Then $\mathrm{ht}(\mathfrak{p})\le n$ ^{[Krull 1937]}.

The proof is by induction on $n$, with Krull's PIT as the base case. The statement also admits a topological interpretation: the codimension of a subvariety cut out by $n$ equations is at most $n$.

Theorem 2 (Dimension of polynomial rings). For any field $k$ and any positive integer $n$, $\dim k[x_1,\dots ,x_n]=n$ ^{[Matsumura 1986]}.

The chain $(0)\subset (x_1)\subset (x_1,x_2)\subset \cdots \subset (x_1,\dots ,x_n)$ shows $\dim k[x_1,\dots ,x_n]\ge n$. The reverse inequality follows from the generalized PIT applied to any maximal ideal (which is generated by $n$ elements after Noether normalisation). More precisely, the Hilbert basis theorem 01.02.17 guarantees that $k[x_1,\dots ,x_n]$ is Noetherian, so the generalized PIT applies to every ideal.

Theorem 3 (Dimension and integral extensions). If $R\subset S$ is an integral extension of commutative rings, then $\dim R=\dim S$.

This follows from the going-up theorem: for every chain ${\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_n$ in $\mathrm{Spec}R$ there exists a chain ${\mathfrak{P}}_0⊊\cdots ⊊{\mathfrak{P}}_n$ in $\mathrm{Spec}S$ with ${\mathfrak{P}}_i\cap R={\mathfrak{p}}_i$; and the lying-over and incomparability properties guarantee the reverse inequality.

Theorem 4 (System of parameters). Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d$. Then $d$ is the minimum number of elements needed to generate an $\mathfrak{m}$-primary ideal, and $d$ equals ${\dim }_{R/\mathfrak{m}}(\mathfrak{m}/{\mathfrak{m}}^2)$ when $R$ is regular.

A set of $d$ elements generating an $\mathfrak{m}$-primary ideal is a system of parameters. Its existence follows from the generalized PIT (for the lower bound) and the prime avoidance lemma (for the upper bound), as established in Exercise 8.

Theorem 5 (Regular local rings). A Noetherian local ring $(R,\mathfrak{m})$ is regular if ${\dim }_{R/\mathfrak{m}}(\mathfrak{m}/{\mathfrak{m}}^2)=\dim R$. Equivalently, $\mathfrak{m}$ is generated by $\dim R$ elements. Every regular local ring is an integral domain ^{[Matsumura 1986]}.

Regularity is the algebraic counterpart of smoothness: the local ring of a smooth point on an algebraic variety is regular. The Zariski tangent space at a point has dimension ${\dim }_{R/\mathfrak{m}}(\mathfrak{m}/{\mathfrak{m}}^2)$, and regularity is the condition that this tangent space has the minimal possible dimension.

Theorem 6 (Serre's characterisation of regularity). For a Noetherian local ring $(R,\mathfrak{m})$, the following are equivalent ^{[Serre 1955]}:

1. $R$ is regular ($\mathfrak{m}$ is generated by $\dim R$ elements).
2. $R$ has finite global dimension equal to $\dim R$.
3. Every $R$-module has finite projective dimension.

This is the foundational result connecting homological algebra to dimension theory. The proof uses the Koszul complex on a system of parameters to build a free resolution of $R/\mathfrak{m}$, and conversely shows that finite global dimension forces $\mathfrak{m}/{\mathfrak{m}}^2$ to have the minimal dimension.

Theorem 7 (Auslander-Buchsbaum formula). If $(R,\mathfrak{m})$ is a Noetherian local ring and $M$ is a finitely generated $R$-module of finite projective dimension, then $$ \operatorname{pd}(M) + \operatorname{depth}(M) = \operatorname{depth}(R) $$ ^{[Auslander-Buchsbaum 1957]}.

When $R$ is regular with $\dim R=d$, setting $M=R/\mathfrak{m}$ (depth $0$, projective dimension $d$) recovers the global dimension $d=\mathrm{depth}(R)$, providing the bridge from the Auslander-Buchsbaum formula to Serre's characterisation.

Synthesis. The foundational reason that Krull dimension governs the prime-spectrum geometry of Noetherian rings is the ascending chain condition, which provides the inductive backbone for all height estimates via symbolic powers and Nakayama's lemma. The central insight is that Krull's principal ideal theorem identifies the height of a minimal prime over a principal ideal with the codimension of the corresponding hypersurface. Putting these together with the dimension formula $\dim k[x_1,\dots ,x_n]=n$, the bridge is between the combinatorial data of prime-ideal chains and the geometric data of variety dimension.

This is exactly the structure that generalises from one equation to $n$ equations via the generalized PIT, and the pattern recurs in Serre's characterisation of regular local rings: a local ring $(R,\mathfrak{m})$ is regular if and only if $\mathfrak{m}$ is generated by $\dim R$ elements, if and only if $R$ has finite global dimension. The Auslander-Buchsbaum theorem deepens the identification by connecting projective dimension with depth, and the bridge is between the homological invariants of the ring and the combinatorial invariants of its prime spectrum.

Full proof set [Master]

Proposition 1 (Generalized principal ideal theorem). Let $R$ be a Noetherian ring, $a_1,\dots ,a_n\in R$, and $\mathfrak{p}$ a prime ideal minimal over $(a_1,\dots ,a_n)$. Then $\mathrm{ht}(\mathfrak{p})\le n$.

Proof. By induction on $n$. The base case $n=1$ is Krull's principal ideal theorem (proved above).

Assume $n\ge 2$ and the result holds for $n-1$. Suppose $\mathrm{ht}(\mathfrak{p})>n$. Choose primes ${\mathfrak{q}}_0⊊\cdots ⊊{\mathfrak{q}}_n⊊\mathfrak{p}$.

Localise at $\mathfrak{p}$: we may assume $(R,\mathfrak{m})$ is a Noetherian local ring with $\mathfrak{m}$ minimal over $(a_1,\dots ,a_n)$, and ${\mathfrak{q}}_0⊊\cdots ⊊{\mathfrak{q}}_n⊊\mathfrak{m}$ is a chain of length $n+1$.

Not all $a_i$ lie in ${\mathfrak{q}}_n$ (otherwise ${\mathfrak{q}}_n$ would contain the ideal $(a_1,\dots ,a_n)$ and $\mathfrak{m}$ would not be minimal over it). Relabel so $a_n\notin {\mathfrak{q}}_n$. Since ${\mathfrak{q}}_i\subset {\mathfrak{q}}_n$ for $i\le n$, we have $a_n\notin {\mathfrak{q}}_i$ for all $i\le n$.

Pass to $\bar{R}=R/(a_n)$. The image $\bar{\mathfrak{m}}$ is minimal over $({\bar{a}}_1,\dots ,{\bar{a}}_{n-1})$, a set of $n-1$ elements. The images ${\bar{\mathfrak{q}}}_0,\dots ,{\bar{\mathfrak{q}}}_n$ are distinct primes (since $a_n\notin {\mathfrak{q}}_i$) with ${\bar{\mathfrak{q}}}_0⊊\cdots ⊊{\bar{\mathfrak{q}}}_n⊊\bar{\mathfrak{m}}$, a chain of length $n+1$ in $\bar{R}$.

By the inductive hypothesis applied to $\bar{\mathfrak{m}}$ and the $n-1$ generators ${\bar{a}}_1,\dots ,{\bar{a}}_{n-1}$: $\mathrm{ht}(\bar{\mathfrak{m}})\le n-1$. But the chain ${\bar{\mathfrak{q}}}_0⊊\cdots ⊊{\bar{\mathfrak{q}}}_n⊊\bar{\mathfrak{m}}$ shows $\mathrm{ht}(\bar{\mathfrak{m}})\ge n+1$. Contradiction. $\square$

Proposition 2 (Dimension of polynomial rings). For any field $k$ and $n\ge 0$, $\dim k[x_1,\dots ,x_n]=n$.

Proof. The chain $(0)\subset (x_1)\subset (x_1,x_2)\subset \cdots \subset (x_1,\dots ,x_n)$ of prime ideals has length $n$, so $\dim k[x_1,\dots ,x_n]\ge n$.

For the reverse inequality, by the Hilbert basis theorem 01.02.17 the ring $k[x_1,\dots ,x_n]$ is Noetherian. Let $\mathfrak{M}$ be any maximal ideal. By the Nullstellensatz (when $k$ is algebraically closed), $\mathfrak{M}=(x_1-a_1,\dots ,x_n-a_n)$ for some $a_i\in k$, generated by $n$ elements. By the generalized PIT (Proposition 1), $\mathrm{ht}(\mathfrak{M})\le n$. Since every chain of primes terminates in a maximal ideal, $\dim k[x_1,\dots ,x_n]\le n$.

For general (not necessarily algebraically closed) fields, the Noether normalisation lemma provides algebraically independent $y_1,\dots ,y_n\in k[x_1,\dots ,x_n]$ such that $k[x_1,\dots ,x_n]$ is integral over $B=k[y_1,\dots ,y_n]$. By the going-up theorem, $\dim k[x_1,\dots ,x_n]=\dim B$. The chain $(0)\subset (y_1)\subset \cdots \subset (y_1,\dots ,y_n)$ in $B$ has length $n$. Every maximal ideal of $B$ has height at most $n$ by the generalized PIT (since $B$ is generated by $n$ elements as a $k$-algebra, every maximal ideal is generated by at most $n$ elements). So $\dim B=n$. $\square$

Connections [Master]

• Hilbert basis theorem and Noetherian rings 01.02.17. The Noetherian hypothesis is essential for Krull's principal ideal theorem: the proof uses the ascending chain condition to stabilise the chain of ideals $(a)+{\mathfrak{q}}^{(n)}$, and Nakayama's lemma requires finite generation. The Hilbert basis theorem guarantees that polynomial rings over Noetherian rings are Noetherian, which is the structural condition that makes the dimension formula $\dim k[x_1,\dots ,x_n]=n$ hold.

• Exact sequences and the snake lemma 01.02.11. The Auslander-Buchsbaum formula relates projective dimension to depth through a long exact sequence in homology, and Serre's characterisation of regular local rings uses projective resolutions. The long exact sequence for $\mathrm{Ext}$ and $\mathrm{Tor}$ modules underpins the depth-sensitivity arguments in the homological theory of local rings.

• Exterior algebra and the Koszul complex 01.02.19. The Koszul complex, built from the exterior algebra on a free module, is the computational tool for studying regular local rings. Given a system of parameters $a_1,\dots ,a_d$ in a local ring $(R,\mathfrak{m})$, the Koszul complex $K_{\bullet }(a_1,\dots ,a_d)$ is a free resolution of $R/(a_1,\dots ,a_d)$ precisely when $R$ is regular. This identifies the exterior-algebra construction with the dimension-theoretic notion of regularity.

Historical & philosophical context [Master]

Krull 1937 introduced the dimension theory of commutative rings in Beitraege zur Arithmetik kommutativer Integritaetsbereiche ^{[Krull 1937]}, proving the principal ideal theorem and establishing the notions of height and dimension for Noetherian rings. The generalised form (arbitrary number of generators) appeared in the same paper. Krull's earlier Idealtheorie (1935) monograph had already laid the groundwork for local ring theory, but the 1937 paper is the origin of the height theorem itself.

Serre 1955 in Faisceaux algebriques coherents ^{[Serre 1955]} introduced the homological characterisation of regular local rings via finite global dimension, establishing the bridge between Krull dimension and homological algebra. The Auslander-Buchsbaum formula ^{[Auslander-Buchsbaum 1957]} refined this connection by relating projective dimension to depth in Noetherian local rings, completing the identification of geometric, algebraic, and homological notions of dimension.

Bibliography [Master]

@article{Krull1937,
  author = {Krull, Wolfgang},
  title = {Beitraege zur Arithmetik kommutativer Integritaetsbereiche},
  journal = {Mathematische Zeitschrift},
  volume = {41},
  year = {1937},
  pages = {665--679},
}

@article{Serre1955,
  author = {Serre, Jean-Pierre},
  title = {Faisceaux algebriques coherents},
  journal = {Annals of Mathematics},
  volume = {61},
  year = {1955},
  pages = {197--278},
}

@article{AuslanderBuchsbaum1957,
  author = {Auslander, Maurice and Buchsbaum, David A.},
  title = {Homological dimension of local rings},
  journal = {Annals of Mathematics},
  volume = {65},
  year = {1957},
  pages = {263--274},
}

@book{AtiyahMacdonald1969,
  author = {Atiyah, Michael F. and Macdonald, Ian G.},
  title = {Introduction to Commutative Algebra},
  publisher = {Addison-Wesley},
  year = {1969},
}

@book{Matsumura1986,
  author = {Matsumura, Hideyuki},
  title = {Commutative Ring Theory},
  publisher = {Cambridge University Press},
  year = {1986},
  series = {Cambridge Studies in Advanced Mathematics 8},
}

@book{Eisenbud1995,
  author = {Eisenbud, David},
  title = {Commutative Algebra with a View Toward Algebraic Geometry},
  publisher = {Springer},
  year = {1995},
  series = {Graduate Texts in Mathematics 150},
}