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Category Archives: Notes
Exact Sequences and the Grothendieck Group
As before, all rings are not commutative in general. Definition. An exact sequence of Rmodules is a collection of Rmodules and a sequence of Rmodule 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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KrullSchmidt Theorem
Here, we will prove that the process of decomposing is unique, given that M is noetherian and artinian. Again, R is a ring, possibly noncommutative. Definition. A decomposition of an Rmodule M is an expression for nonzero modules An Rmodule M is said … Continue reading
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Tagged indecomposable modules, krullschmidt, local rings, matrix rings, splitting lemma, unique factorisation
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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
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Tagged algebra, associative algebra, indecomposable modules, local rings, units
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Jacabson Radical
Recall that the radical of the base ring R is called its Jacobson radical and denoted by J(R); this is a twosided ideal of R. Earlier, we had proven that a ring R is semisimple if and only if it is artinian and J(R) = … Continue reading
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Tagged artinian, hopkinslevitzki, jacobson radical, matrix rings, nilpotent ideals, noetherian, semisimple rings
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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 finitedimensional 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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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
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Tagged algebra, artinian, noetherian, semisimple rings, simple modules
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The Group Algebra (III)
As alluded to at the end of the previous article, we shall consider the case where K is algebraically closed, i.e. every polynomial with coefficients in K factors as a product of linear polynomials. E.g. K = C is a common choice. Having assumed … Continue reading
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Tagged character theory, division rings, group algebras, quaternions, semisimple rings, simple modules
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The Group Algebra (II)
We continue our discussion of the group algebra. Constructing K[G]modules Recall that such a module V is also called a representation of G over K, and corresponds to a group homomorphism (i) Given a K[G]module V, a submodule W of V is precisely a vector subspace W such that g(W) ⊆ W for … Continue reading
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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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