Tag Archives: algebra

Local Rings

Mathematicians are generally more familiar with the case of local commutative rings, so we’ll begin from there. Definition. A commutative ring R is said to be local if it has a unique maximal ideal. Note that every non-zero commutative ring has … Continue reading

Posted in Notes | Tagged , , , , | Leave a comment

Composition Series

Positive integers can be uniquely factored as a product of primes. Here, we would like to prove a counterpart for modules. Now there are two ways to “factor” a module M; a more liberal way takes a submodule N which gives us composition … Continue reading

Posted in Notes | Tagged , , , , , , , | Leave a comment

Radical of Module

As mentioned in the previous article, we will now describe the “bad elements” in a ring R which stops it from being semisimple. Consider the following ring: Since R is finite-dimensional over the reals, it is both artinian and noetherian. However, R is not … Continue reading

Posted in Notes | Tagged , , , , , , | Leave a comment

Noetherian and Artinian Rings and Modules

We saw the case of the semisimple ring R, which is a (direct) sum of its simple left ideals. Such a ring turned out to be nothing more than a finite product of matrix algebras. One asks if there is a … Continue reading

Posted in Notes | Tagged , , , , | Leave a comment

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 , , , , , | 2 Comments

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 , , , , | Leave a comment

Topology: More on Algebra and Topology

We’ve arrived at the domain where topology meets algebra. Thus we have to proceed carefully to ensure that the topology of our algebraic constructions are well-behaved. Let’s look at topological groups again. Our first task is to show that the … Continue reading

Posted in Notes | Tagged , , , , , , , , | Leave a comment