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For a prime ideal $P$ of a polynomial ring over a field, the quotient's Krull dimension is the transcendence degree of its fraction field, and its height is $n$ minus that degree. This dimension theorem identifies algebraic codimension with the length of prime chains. It must not be applied indiscriminately to every Noetherian ring without its polynomial-ring hypotheses.

A principal fractional ideal of a number field is $a{\mathcal{O}}_K$ for a nonzero $a\in K$. Its generators differ by a unit in the ring of integers of a number field. Quotienting nonzero fractional ideals by principal fractional ideals gives the ideal class group; requiring congruences and signs of the generators instead gives a ray class group.

For a normal domain $A$ with fraction field $K$, the field trace of an integral element in a finite separable field extension $L/K$ belongs to $A$. Its conjugates are integral, so their sum is integral; it lies in $K$ and hence in $A$ by normality. This places the integral closure inside a trace-dual lattice.

A uniformizer of a discrete valuation ring $R$ is a generator $\pi$ of its unique nonzero maximal ideal. Every nonzero $x$ in the fraction field has a unique expression $x=u{\pi }^n$ with $u$ a unit and $n\in \mathbb{Z}$.

Every nonzero nonunit in a unique factorization domain is a finite product of irreducible elements. Such a factorization is unique up to permutation and multiplication of the factors by units. Irreducibles are prime elements, so divisibility can be checked by comparing their multiplicities. This applies to a polynomial ring over a field and is useful for intersections of localizations inside its fraction field.

Every local subring $A$ of a field $K$ is dominated by a valuation ring with fraction field $K$. Order local overrings by inclusion and contraction of their maximal ideals; chain unions are local, so Zorn's lemma gives a maximal one. For each $z\in K^{\times }$, at least one of the extensions of its maximal ideal to $V[z]$ and $V[z^{-1}]$ is proper: otherwise two least-degree relations for $1$ with maximal-ideal coefficients reduce one another to a smaller-degree relation. Localizing the proper extension gives a dominating overring, so maximality forces either $z$ or $z^{-1}$ into $V$.

A valuation ring is an integral domain $R$ such that, for every nonzero element $x$ of its fraction field $K$, either $x\in R$ or $x^{-1}\in R$. Equivalently, its ideals are totally ordered by inclusion.

If an element $x$ of the fraction field is absent from a valuation ring $V$, its inverse $y$ belongs to the maximal ideal. A monic equation for $x$, multiplied by a power of $y$, would put $1$ in that maximal ideal. Thus $V$ is an integrally closed domain, without a discrete or Noetherian hypothesis.

For a finite-type morphism of Noetherian schemes, separatedness is equivalent to uniqueness in every lifting problem $\mathrm{Spec}K\to X$ over $\mathrm{Spec}R\to Y$, where $R$ is a valuation ring with fraction field $K$.