Course - Rings and Modules HT24
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.
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- Lecture Notes
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”$\mathbb F[t]$-modules are just $\mathbb F$-vector spaces equipped with an endomorphism”!
Notes
- [[Notes - Rings and Modules HT24, Overview of results and relationships in rings]]U
- [[Notes - Rings and Modules HT24, Basic definitions for modules]]U
- [[Notes - Rings and Modules HT24, Basic definitions for rings]]U
- [[Notes - Rings and Modules HT24, Chinese remainder theorem]]U
- [[Notes - Rings and Modules HT24, Correspondence theorems]]U
- [[Notes - Rings and Modules HT24, Divisibility]]U
- [[Notes - Rings and Modules HT24, Euclidean domains]]U
- [[Notes - Rings and Modules HT24, Factorisation in polynomial rings]]U
- [[Notes - Rings and Modules HT24, Factorisation]]U
- [[Notes - Rings and Modules HT24, Fields]]U
- [[Notes - Rings and Modules HT24, Free modules]]U
- [[Notes - Rings and Modules HT24, Ideals]]U
- [[Notes - Rings and Modules HT24, Integral domains]]U
- [[Notes - Rings and Modules HT24, Isomorphism theorems]]U
- [[Notes - Rings and Modules HT24, Matrices over a ring]]U
- [[Notes - Rings and Modules HT24, Polynomial rings]]U
- [[Notes - Rings and Modules HT24, Presentations]]U
- [[Notes - Rings and Modules HT24, Prime and maximal ideals]]U
- [[Notes - Rings and Modules HT24, Principal ideal domains]]U
- [[Notes - Rings and Modules HT24, Ring quotients]]U
- [[Notes - Rings and Modules HT24, Smith normal form]]U
- [[Notes - Rings and Modules HT24, Structure theorems]]U
- [[Notes - Rings and Modules HT24, Torsion]]U