Modules over PIDs

$\def\R{\mathbb{R}} \def\C{\mathbb{C}} \def\Q{\mathbb{Q}} \def\N{\mathbb{N}} \def\Z{\mathbb{Z}} \def\mfa{\mathfrak{a}} \def\mfb{\mathfrak{b}} \def\mfc{\mathfrak{c}} \def\mfp{\mathfrak{p}} \def\mfq{\mathfrak{q}} \def\mfm{\mathfrak{m}} \def\ev{\mathrm{ev}} \def\Im{\mathrm{Im}} \def\End{\mathrm{End}} \def\Aut{\mathrm{Aut}} \def\Int{\mathrm{Int}} \def\Hom{\mathrm{Hom}} \def\Iso{\mathrm{Iso}} \def\GL{\mathrm{GL}} \def\ker{\mathrm{ker }} \def\Span{\mathrm{Span}} \def\tr{\mathrm{tr}} \def\Rings{\mathrm{Rings}} \def\Alg{{F\text{-Alg}}} \def\Ann{{\rm Ann}} \newcommand{\unit}{1\!\!1} \usepackage{tikz} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} $

Chapter 10 Modules over PIDs

[To be written]

We have followed the first two sections from Chapter 12; ‘Abstract Algebra’ by Dummit and Foote.

For computational verification you may use the following SageMath calculator to calculate rational/Jordan canonical form. In the following SageMath source the matrix is written row-wise and over $\Q$ (written as QQ in the SageMath code). We can change the row-entries to get the canonical forms.

Rational Canonical Form

Jordan Canonical Form: We refer to Linear Algebra notes ¹ for more details.

linear-algebra.netlify.app/sec-jordan-normal-form

Algebra-II: Companion notes

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Chapter 10 Modules over PIDs