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Tag Archives: modules
Hom Functor
Fret not if you’re unfamiliar with the term functor; it’s a concept in category theory we will use implicitly without delving into the specific definition. This topic is, unfortunately, a little on the dry side but it’s a necessary evil to get … Continue reading
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Tagged bimodules, hom functor, left modules, left-exact, modules, right modules
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Exact Sequences and the Grothendieck Group
As before, all rings are not commutative in general. Definition. An exact sequence of R-modules is a collection of R-modules and a sequence of R-module homomorphisms: such that for all i. Examples 1. The sequence is exact if and only if f … Continue reading
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Tagged composition series, exact sequences, grothendieck group, modules, short exact sequences
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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
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Tagged algebra, artinian, composition series, length of module, matrix rings, modules, noetherian, unique factorisation
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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
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Tagged algebra, artinian, jacobson radical, matrix rings, modules, radical of modules, semisimple rings
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The Group Algebra (I)
[ Note: the contents of this article overlap with a previous series on character theory. ] Let K be a field and G a finite group. The group algebra K[G] is defined to be a vector space over K with basis , where “g” here is … Continue reading
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Tagged character theory, group actions, group algebras, modules, representation theory, semisimple rings, simple modules
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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
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Tagged cokernels, direct products, direct sums, homomorphism, isomorphism theorems, kernels, linear algebra, modules, submodules, vector space
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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
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Tagged cokernels, direct products, direct sums, homomorphism, isomorphism theorems, kernels, modules, submodules
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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
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Tagged generated submodules, ideals, left ideals, modules, rings, scalar multiplication, simple modules, submodules
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Quick Guide to Character Theory (I): Foundation
Character theory is one of the most beautiful topics in undergraduate mathematics; the objective is to study the structure of a finite group G by letting it act on vector spaces. Earlier, we had already seen some interesting results (e.g. proof … Continue reading
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Tagged character theory, dual spaces, fields, group algebras, groups, modules, representation theory
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