Category Archives: Notes

Exact Sequences and the Grothendieck Group

Posted on January 20, 2015 by limsup

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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Krull-Schmidt Theorem

Posted on January 17, 2015 by limsup

Here, we will prove that the process of decomposing is unique, given that M is noetherian and artinian. Again, R is a ring, possibly non-commutative. Definition. A decomposition of an R-module M is an expression for non-zero modules An R-module M is said … Continue reading →

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Local Rings

Posted on January 15, 2015 by limsup

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 →

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Jacabson Radical

Posted on January 13, 2015 by limsup

Recall that the radical of the base ring R is called its Jacobson radical and denoted by J(R); this is a two-sided 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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Composition Series

Posted on January 10, 2015 by limsup

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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Radical of Module

Posted on January 9, 2015 by limsup

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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Noetherian and Artinian Rings and Modules

Posted on January 8, 2015 by limsup

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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The Group Algebra (III)

Posted on January 6, 2015 by limsup

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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The Group Algebra (II)

Posted on January 5, 2015 by limsup

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)

Posted on January 3, 2015 by limsup

[ 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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