DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Determinantal divisors from the minors of a matrix over a PID

Definition

Let $A \in M_{m \times n} (R)$ over a principal ideal domain $R$ . For $k \geq 1$ , the $k$ -th determinantal ideal $D_{k} (A)$ is the ideal generated by the determinants of all $k \times k$ minors of $A$ (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix, The ideal generated by a subset and principal ideals). Set $D_{0} (A) = R$ . If $k > \min (m, n)$ , there are no such minors and $D_{k} (A) = (0)$ .

Since $R$ is a PID (Principal ideal domain), write $D_{k} (A) = (Δ_{k} (A))$ . The associate class of $Δ_{k} (A)$ is the $k$ -th determinantal divisor. Thus $Δ_{0} (A)$ is a unit up to associates, while a vanishing determinantal ideal has divisor $0$ .

Depends on

Used by

Smith normal form is unique through the gcds of its minors Theorem

Dependency tree · two levels

15 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

M. Brussel, Finitely Generated Modules over a PID, Section 2.1 (standard reference, not scraped)
A. Apisa, Wisconsin Math 542, Lectures 8-9 (standard reference, not scraped)