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Tag Archives: algebra
Commutative Algebra 18
Basics of Category Theory As we proceed, we should cover some rudimentary category theory or many of the subsequent constructions would seem unmotivated. The essence of category is in studying algebraic objects and the homomorphisms between them. By now we have … Continue reading
Posted in Advanced Algebra
Tagged algebra, category theory, contravariant, coslice categories, covariant, functors, morphisms, rings
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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 nonzero commutative ring has … Continue reading
Posted in Notes
Tagged algebra, associative algebra, indecomposable modules, local rings, units
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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
Posted in Notes
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 finitedimensional over the reals, it is both artinian and noetherian. However, R is not … Continue reading
Posted in Notes
Tagged algebra, artinian, jacobson radical, matrix rings, modules, radical of modules, semisimple rings
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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 algebra, artinian, noetherian, semisimple rings, simple modules
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Simple Modules
We briefly talked about modules over a (possibly noncommutative) 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 Zmodule given by Z/2Z. [ Recall: a Zmodule is … Continue reading
Posted in Notes
Tagged algebra, associative algebra, division rings, rings, schur's lemma, simple modules
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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
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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 wellbehaved. Let’s look at topological groups again. Our first task is to show that the … Continue reading
Posted in Notes
Tagged algebra, connected spaces, groups, isomorphism theorems, open maps, orthogonal groups, quotient topology, rings, topology
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Symmetric Polynomials (III)
Now we generalise this to n variables: . It’s clear what the corresponding building blocks of symmetric polynomials would be: ; ; ; … . We call these ei‘s the elementary symmetric polynomials in the xi‘s. Note that each ei is the coefficient of Ti in the … Continue reading
Symmetric Polynomials (II)
When we move on to n=3 variables, we now have, as basic building blocks, These are just the coefficients of in the expansion of . Once again, any symmetric polynomial in x, y, z with integer coefficients can be expressed as a polynomial … Continue reading