-
Recent Posts
Archives
- March 2023
- January 2023
- May 2020
- April 2020
- March 2020
- June 2018
- July 2016
- June 2016
- May 2016
- March 2015
- February 2015
- January 2015
- December 2014
- December 2013
- November 2013
- July 2013
- June 2013
- May 2013
- March 2013
- February 2013
- January 2013
- December 2012
- November 2012
- October 2012
- September 2012
- August 2012
- April 2012
- March 2012
- February 2012
- January 2012
- December 2011
- November 2011
- October 2011
Categories
Meta
Pages
Category Archives: Notes
Structure of Semisimple Rings
It turns out there is a nice classification for semisimple rings. Theorem. Any semisimple ring R is a finite product: where each is a division ring and is the ring of n × n matrices with entries in D. Furthermore, the … Continue reading
Semisimple Rings and Modules
After discussing simple modules, the next best thing is to look at semisimple modules, which are just direct sums of simple modules. Here’s a summary of the results we’ll prove: A module is semisimple iff it is a sum of simple … Continue reading
Simple Modules
We briefly talked about modules over a (possibly non-commutative) ring R. An important aspect of modules is that unlike vector spaces, modules are usually not free, i.e. they don’t have a basis. For example, take the Z-module given by Z/2Z. [ Recall: a Z-module is … Continue reading
Posted in Notes
Tagged algebra, associative algebra, division rings, rings, schur's lemma, simple modules
2 Comments
From Euler Characteristics to Cohomology (II)
Boundary Maps Here’s a brief recap of the previous article: we learnt that in refining a cell decomposition of an object M, we can, at each step, pick an i-dimensional cell and divide it in two. In this way, we … Continue reading
Posted in Notes
Tagged advanced, betti numbers, cell complexes, cellular homology, euler characteristics, homology, simplicial complexes, topology
2 Comments
From Euler Characteristics to Cohomology (I)
[ Warning: this is primarily an expository article, so the proofs are not airtight, but they should be sufficiently convincing. ] The five platonic solids were well-known among the ancient Greeks (V, E, F denote the number of vertices, edges and faces respectively): … Continue reading
Elementary Module Theory (IV): Linear Algebra
Throughout this article, a general ring is denoted R while a division ring is denoted D. Dimension of a Vector Space First, let’s consider the dimension of a vector space V over D, denoted dim(V). If W is a subspace of V, we proved earlier that … Continue reading
Posted in Notes
Tagged advanced, algebra, linear algebra, module homomorphism, vector spaces
Leave a comment
Elementary Module Theory (III): Approaching Linear Algebra
The Hom Group Continuing from the previous installation, here’s another way of writing the universal properties for direct sums and products. Let Hom(M, N) be the set of all module homomorphisms M → N; then: (*) for any R-module N. In the case where there’re finitely … Continue reading
Posted in Notes
Tagged cokernels, direct products, direct sums, homomorphism, isomorphism theorems, kernels, linear algebra, modules, submodules, vector space
Leave a comment
Elementary Module Theory (II)
Having defined submodules, let’s proceed to quotient modules. Unlike the case of groups and rings, any submodule can give a quotient module without any additional condition imposed. Definition. Let N be a submodule of M. By definition, it’s an additive … Continue reading
Posted in Notes
Tagged cokernels, direct products, direct sums, homomorphism, isomorphism theorems, kernels, modules, submodules
Leave a comment
Elementary Module Theory (I)
Modules can be likened to “vector spaces for rings”. To be specific, we shall see later that a vector space is precisely a module over a field (or in some cases, a division ring). This set of notes assumes the … Continue reading
Posted in Notes
Tagged generated submodules, ideals, left ideals, modules, rings, scalar multiplication, simple modules, submodules
Leave a comment
Quick Guide to Character Theory (III): Examples and Further Topics
G10(a). Character Table of S4 Let’s construct the character table for . First, we have the trivial and alternating representations (see examples 1 and 2 in G1), both of which are clearly irreducible. Next, the action of G on {1, 2, 3, … Continue reading
Mathematics and Such 