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An affine algebraic curve is an affine variety of Krull dimension one. An irreducible affine curve has a finitely generated one-dimensional domain as its coordinate ring. Quotienting this ring by a nonzero element gives a zero-dimensional Noetherian ring, hence an Artinian ring. Consequently a nonzero regular function has finitely many zeros. No nonsingularity assumption is part of the definition.

An Artinian ring satisfies the descending chain condition on ideals. A commutative ring is Artinian exactly when it is Noetherian and has Krull dimension zero.

A commutative Noetherian ring is Cohen-Macaulay when every localization at a prime ideal has depth of a Noetherian local ring equal to its Krull dimension. A regular local ring is Cohen-Macaulay. If $(R,\mathfrak{m})$ is a Cohen-Macaulay local ring and $x\in \mathfrak{m}$ is a non-zero-divisor, then $R/(x)$ is Cohen-Macaulay and both its depth of a Noetherian local ring and Krull dimension are one less than those of $R$.

The requirement $x\in \mathfrak{m}$ excludes units, whose quotient would be the zero ring. In particular, a hypersurface in a regular ambient variety satisfies the depth condition $(S_2)$ in the Serre criterion for normality.

For a local ring $(R,\mathfrak{m})$ with residue field $\kappa =R/\mathfrak{m}$, its cotangent space is the $\kappa$-vector space $\mathfrak{m}/{\mathfrak{m}}^2$. In a Noetherian local ring, its dimension is the minimal number of generators of $\mathfrak{m}$, by the Nakayama lemma. Equality of this dimension with the Krull dimension defines a regular local ring.

A Dedekind domain is a Noetherian ring that is an integrally closed domain of Krull dimension one. Equivalently, for a Noetherian domain that is not a field, every localization at a prime ideal at a nonzero prime is a discrete valuation ring. The Noetherian and nonfield qualifications belong to this local characterization; a field would satisfy its condition vacuously.

The ring of integers of a number field is a standard example. Every nonzero proper ideal has a unique factorization into nonzero prime ideals, even when element factorization is not unique. Locally, the exponent of each prime is the discrete valuation of that ideal in its corresponding discrete valuation ring. This connects unique factorization of ideals in a Dedekind domain with local arithmetic.

The depth of a Noetherian local ring is the maximal length of an $R$-regular sequence in its maximal ideal. It is at most its Krull dimension. Quotienting by a nonzero divisor in the maximal ideal lowers depth by one. A unit is excluded from this assertion, since its quotient is the zero ring. This invariant measures how many independent successive nonzero-divisor equations the local algebra supports.

For a nonempty irreducible affine variety $X$ over an algebraically closed field,Every closed-point local ring has Krull dimension $\dim X$, and its embedding dimension is at least that value. Equality holds precisely when the local ring is regular, hence at a smooth point of a variety. The density of the smooth locus ensures the minimum is attained. The same statement holds for a pure-dimensional reduced variety; components of different dimensions require local dimension instead.^[ref]

For example, $X=V(xz,yz)\subset {\mathbb{A}}^3$ is the union of the plane $z=0$ and the vertical line $x=y=0$. It has dimension two, but its Zariski tangent space at $(0,0,1)$ has dimension one. That point is smooth on its one-dimensional component. Thus neither the global minimum formula nor testing all points against one global dimension works for a mixed-dimensional variety.

If $f\in k[Y][X]$ is monic of positive degree in $X$, monic division makes $k[Y][X]/(f)$ free of rank ${\mathrm{deg}}_{X}f$ over $k[Y]$. The inclusion of $k[Y]$ is injective and the quotient is an integral extension, so its Krull dimension is one. This argument applies even when the polynomial factors or the field has positive characteristic.

The supremum of the lengths of strict chains of nonempty irreducible closed subsets. It is the maximum of the dimensions of the irreducible components. For an affine algebraic set it equals the Krull dimension of its coordinate ring; for a quasi-projective algebraic set it is the supremum of these dimensions over affine open subsets.

The dimension of a scheme is the supremum of the lengths of strict chains of irreducible closed subsets of its underlying topological space. On an affine scheme it is the Krull dimension of its coordinate ring. Taking the reduced subscheme leaves the dimension unchanged.

For a nonzero finitely generated module over a complex Weyl algebra, choose a good filtration for the order filtration and take the Krull dimension of its support of a module over $\mathrm{gr}R=\mathbb{C}[x_1,\dots ,x_n,{\xi }_1,\dots ,{\xi }_n]$. This is the dimension of its order characteristic variety of a Weyl algebra module. One may set $d(0)=-\infty$.

For independence from the good module filtration, any two such filtrations of a module are bounded shifts of one another, by their finite shifted-generating descriptions. If a homogeneous symbol $\sigma (a)$ of degree $d$ annihilates one associated graded module, then $a{F}_{j}M\subseteq F_{j+d-1}M$ and its $q$-th power lowers the expected degree by $q$. For shift bound $c$ and $q>2c$, comparison shows that $\sigma (a{)}^q$ annihilates the other associated graded module. Reverse the roles to obtain equality of the radicals of ideals of their annihilators, hence equality of their support of a module and Krull dimension. This argument holds the algebra filtration fixed.

The embedding dimension of a Noetherian local ring $(R,\mathfrak{m})$ is ${\dim }_{R/\mathfrak{m}}(\mathfrak{m}/{\mathfrak{m}}^2)$. It counts the smallest number of local generators needed for the maximal ideal. It is at least the Krull dimension; equality characterizes a regular local ring.

Let $(A,\mathfrak{m})$ be a regular local ring of Krull dimension $d$. Its formal power series ring $B=A[[x]]$ is Noetherian and local, with maximal ideal $I=\mathfrak{m}B+(x)$ and embedding dimension $d+1$. Since $I$ is generated by $d+1$ elements, the Krull height theorem gives $\dim B\le d+1$. For the reverse inequality choose a strict prime chain ${\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_d=\mathfrak{m}$ in $A$. Finite generation of each ${\mathfrak{p}}_i$ gives $B/{\mathfrak{p}}_{i}B\cong (A/{\mathfrak{p}}_i)[[x]]$, an integral domain, so each extended ideal is prime. The inclusions remain strict by contraction to $A$, and $\mathfrak{m}B⊊I$ adds one more step since $x\notin \mathfrak{m}B$. This proves $\dim B=d+1$, equal to its embedding dimension, and hence regularity.

Let a commutative $k$-algebra $R$ be generated by $x_1,\dots ,x_n$, and let $R_j$ be the span of words of total degree at most $j$ in those generators. The associated graded ring of this filtration is a standard graded algebra, andThe Hilbert-Serre theorem therefore makes ${\dim }_k{R}_j$ eventually a polynomial in $j$. Its degree is $\dim R$; this follows by applying the Rees-ring deformation, whose special fibers are $R$ and the associated graded ring, so they have the same Krull dimension.

Height measures prime-chain length within Krull dimension theory.

The height of an ideal $I$ is the infimum of the heights of prime ideals containing it:The height of a prime is the supremum of the lengths of strict prime chains ending at that prime.

The height of a prime ideal is the supremum of lengths of strict prime chains ending at it, equivalently the Krull dimension of its localization at a prime ideal. In a Noetherian ring, the Krull height theorem bounds the height of a minimal prime over an $n$-generated ideal by $n$.

If $M$ has Krull dimension $d$, the coefficient of $n^d$ in its Hilbert-Samuel polynomial is $e_I(M)/d!$. The positive integer $e_I(M)$ is the Hilbert–Samuel multiplicity of $M$ with respect to $I$.

An ideal $I$ of a Noetherian local ring $(A,\mathfrak{m})$ is an ideal of definition when $A/I$ has Krull dimension zero. Equivalently, $I$ is $\mathfrak{m}$-primary, so $\sqrt{I}=\mathfrak{m}$ and ${\mathfrak{m}}^r\subseteq I$ for some positive integer $r$.

If a Noetherian ring has finite Krull dimension $d$, then it has infinitely many prime ideals of each height $n$ with $0<n<d$. In a prime chain of maximal length, apply prime ideals between a three-prime chain to the terms of heights $n-1$, $n$, and $n+1$.

The Krull dimension of a commutative ring $R$ is the supremum of the lengths $n$ of strict chains of prime ideals ${\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_n$. The length counts inclusions, not the number of ideals.

A field has dimension zero. The integers have dimension one, with chains $(0)⊊(p)$, and $k[X_1,\dots ,X_n]$ over a field has dimension $n$. In algebraic geometry, chains of primes correspond to nested irreducible closed subsets, so this algebraic definition captures geometric dimension. It is distinct from the dimension of a vector space.