Principal ideal domain

Depends on

• Zero divisor, and integral domain: a commutative ring with $1 \ne 0$ and no zero divisors
• The ideal generated by a subset and principal ideals

Used by

• Every algebra of finite type over a principal ideal domain is a Noetherian ring Corollary
• Every DVR is a PID Corollary
• Every principal ideal domain is Noetherian Corollary
• For every field F, F[x] is a principal ideal domain Corollary
• The fundamental theorem of finitely generated abelian groups from PID modules Corollary
• The index of a full-rank subgroup of ℤⁿ is the absolute determinant of a generating matrix Corollary
• Determinantal divisors from the minors of a matrix over a PID Definition
• Matrix equivalence and Smith normal form over a PID Definition
• Every nonfield PID is a Dedekind domain with trivial class group Example
• Localizing a PID at a nonzero prime Example
• FALSE: every torsion-free module over a PID is free False statement
• Coprime cyclic quotients over a PID split by the Chinese remainder map Lemma
• Every irreducible element of a principal ideal domain is prime Lemma
• The cycle-boundary short exact sequences for a free complex over a PID Lemma
• Under Choice, a submodule of an arbitrary-rank free module over a PID is free Lemma
• A Dedekind domain is a PID exactly when its class group is trivial Theorem
• A submodule of an arbitrary-rank free module over a PID is free Theorem
• Equivalent characterizations of a DVR Theorem
• Euler-Poincare formula for finite free complexes Theorem
• Every Euclidean domain is a principal ideal domain Theorem
• Over a PID, injective modules are exactly divisible modules Theorem
• Over a principal ideal domain flatness is equivalent to torsion-freeness Theorem

Dependency tree · two levels

7 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

• Sharifi, Abstract Algebra, Advanced Ring Theory (standard reference, not scraped)