Course - Rings and Modules HT24

Created: January 11, 2024 | Updated: September 30, 2025 | Read markdown | About these notes | View in context | Study these flashcards

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.
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.
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.

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

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