Matrices over Principal Ideal Domains

HEGL Illustrating Mathematics Seminar
Winter Semester 2024-2025

Vincent Voß
Heidelberg University

Principal Ideal Domain

• Principal Ideal: such that for some a in
• PID: Bézout Domain and satisfies divisor chain condition
  • Bézout´s Identity:
• Examples:

Matrices over PID

• A matrix is diagonal, if for
• Two matrices A and B are equivalent, if there are invertible and such that

Lemma 1.1

Each Matrix over a PID is equivalent to a diagonal Matrix.

Smith Normal Form

Def: A matrix is in SNF if it is diagonal and

Theorem 1.2: Each matrix over a PID is equivalent to a Matrix in SNF

Proof: By Lemma 1.1 we can start with a diagonal matrix

• Corner element divides all of the remaining elements
• repeat for the size of

Example

Lemma 1.3

Def: is the ideal generated by the determinants of all submatrices of

Lemma: and equivalent matrices over a GCD Domain for all

Proof: Let be invertible matrices

• We find that and

Theorem 1.4

Two matrices in SNF over a GCD Domain are equivalent if and only if corresponding elements are associates.

Proof: Let be matrices in SNF. We find that the are determined by its diagonal elements, apart from units

• if and only if the diagonal elements are associates

Finitely represented modules

HEGL Illustrating Mathematics Seminar
Winter Semester 2024-2025

Felix Joeken
Heidelberg University

Misc

For the whole talk is a comm. Ring and is a -Module

Def 0 (from classic mathematics):[^1]

Let be a comm. ring. And be an -module.
M is a finitely presented -module if there exists an exact sequence:

Example 1

Choose

and

Def 1 : finite presentation of an R-Module

Let be a finitely presented Module. A finite presentation of is a triple with generator of and for each

Theorem 1 : Basis change of finite presentation of an R-Module

Let be a finitely presented module and let be an invertible matrix over R. Then

[^1]: The Stacks project. (2024, November 25). https://stacks.math.columbia.edu/tag/0518