# Course - Rings and Modules HT24

> Source: https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/ · Updated: 2025-09-30 · Tags: uni, course

> Rings are algebraic structures where the objects behave like the integers: you can add elements ($a + b$), subtract elements ($a - b$) and multiply elements ($ab$), but multiplicative inverses don't have to exist (e.g. for the integers, $1/2 \notin \mathbb Z$). But unlike the integers, multiplication doesn't have to be commutative. In this way, they generalise fields.
> <br>
> Modules are like vector spaces where instead of having a field of scalars, you have a ring of scalars. This ends up making them more complicated than vector spaces, e.g. it is not true that any linearly independent set can be extended to a basis.
> <br>
> The course ends with a big theorem called the "structure theorem for finitely generated modules over a Euclidean domain", which provides a canonical form for lots of modules.

- [Course Webpage](https://courses.maths.ox.ac.uk/course/view.php?id=4966)
- [Lecture Notes](https://courses.maths.ox.ac.uk/pluginfile.php/93943/mod_resource/content/9/A3pdfLaTeX.pdf)
- Other courses this term: [Courses HT24](https://ollybritton.com/notes/uni/part-a/ht24/courses/)

> "$\mathbb F[t]$-modules are just $\mathbb F$-vector spaces equipped with an endomorphism"!

### Notes
- [Notes - Rings and Modules HT24, IDs, PIDs and EDs hierarchy](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/ids-pids-and-eds-hierarchy/)
- [Notes - Rings and Modules HT24, Modules](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/modules/)
- [Notes - Rings and Modules HT24, Basic definitions](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/basic-definitions/)
- [Notes - Rings and Modules HT24, Chinese remainder theorem](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/chinese-remainder-theorem/)
- [Notes - Rings and Modules HT24, Correspondence theorems](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/correspondence-theorems/)
- [Notes - Rings and Modules HT24, Divisibility](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/divisibility/)
- [Notes - Rings and Modules HT24, Euclidean domains](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/euclidean-domains/)
- [Notes - Rings and Modules HT24, Eisenstein's criterion](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/eisensteins-criterion/)
- [Notes - Rings and Modules HT24, Factorisation in polynomial rings](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/factorisation-in-polynomial-rings/)
- [Notes - Rings and Modules HT24, Unique factorisation](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/unique-factorisation/)
- [Notes - Rings and Modules HT24, Fields](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/fields/)
- [Notes - Rings and Modules HT24, Free modules](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/free-modules/)
- [Notes - Rings and Modules HT24, Ideals](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/ideals/)
- [Notes - Rings and Modules HT24, Integral domains](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/integral-domains/)
- [Notes - Rings and Modules HT24, Isomorphism theorems](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/isomorphism-theorems/)
- [Notes - Rings and Modules HT24, Matrices over a ring](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/matrices-over-a-ring/)
- [Notes - Rings and Modules HT24, Polynomial rings](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/polynomial-rings/)
- [Notes - Rings and Modules HT24, Presentations](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/presentations/)
- [Notes - Rings and Modules HT24, Prime and maximal ideals](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/prime-and-maximal-ideals/)
- [Notes - Rings and Modules HT24, Principal ideal domains](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/principal-ideal-domains/)
- [Notes - Rings and Moudles HT24, Rational and Jordan canonical forms](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/rational-and-jordan-canonical-forms/)
- [Notes - Rings and Modules HT24, Quotients](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/quotients/)
- [Notes - Rings and Modules HT24, Smith normal form](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/smith-normal-form/)
- [Notes - Rings and Modules HT24, Structure theorems](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/structure-theorems/)
- [Notes - Rings and Modules HT24, Torsion](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/notes/torsion/)

### Problem Sheets
- [Sheet 1](https://courses.maths.ox.ac.uk/pluginfile.php/93944/mod_assign/introattachment/0/quest1A3.pdf)
- [Sheet 2](https://courses.maths.ox.ac.uk/pluginfile.php/93945/mod_assign/introattachment/0/quest2A3.pdf)
- [Sheet 3](https://courses.maths.ox.ac.uk/pluginfile.php/93946/mod_assign/introattachment/0/quest3A3.pdf)
- [Sheet 4](https://courses.maths.ox.ac.uk/pluginfile.php/93947/mod_assign/introattachment/0/quest4A3.pdf)

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