Integral elements over a commutative ring and algebraic integers

Depends on

• Algebras over a commutative ring, central structure maps, and algebra homomorphisms
• Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree
• Evaluation and roots of a polynomial in a commutative target ring

Used by

• Integral elements over a nonzero base ring form a subring Corollary
• The Artin–Tate lemma with integrality in place of module finiteness Corollary
• The rational algebraic integers are exactly the integers Corollary
• Quasi-finite does not imply finite Counterexample
• Zₚ is not the integral closure of Z in Qₚ Counterexample
• Integral elements subalgebra of an arbitrary ring map Definition
• Integral ring maps and integral extensions Definition
• Strong transcendence over a subring Definition
• √2 and (1+√5)/2 are algebraic integers, while 1/2 is not Example
• A finite algebra is its own Zariski Main factor Example
• An irreducible curve can have arbitrarily large tangent dimension Example
• Normalization of a nodal affine plane curve Example
• Normalization of the cusp semigroup ring Example
• Normalization of the t³,t⁴,t⁵ monomial curve Example
• The punctured affine line as an open finite factorization Example
• A finite-type field reduces to a localization over a transcendence basis Lemma
• A local domain has a dominating valuation overring Lemma
• A subalgebra generated by finitely many integral elements is module-finite Lemma
• An algebraic-integer average of roots of unity is either 0 or a common root of unity Lemma
• Finite-variable polynomial algebras over fields are integrally closed Lemma
• For a finite group of ring automorphisms the orbit polynomial is monic over the invariant subring, so the ring is integral over its invariants Lemma
• Integral closure in a purely inseparable rational envelope is finite Lemma
• Integral closure is unchanged across an integral intermediate domain Lemma
• One-variable integral correction after leading-coefficient localization Lemma
• Polynomial rings over normal domains are normal Lemma
• The leading coefficient times a root is integral Lemma
• A finite-type domain over a field has finite normalization Theorem
• Finite morphisms are integral and universally closed Theorem
• Integrality and finite-module characterizations for one element Theorem
• Polynomial algebras over fields have finite integral closures Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Eloisa Grifo, Commutative Algebra I, Section 1.4 (standard reference, not scraped)