Monthly Archives: January 2015

Tensor Product over Noncommutative Rings

Posted on January 31, 2015 by limsup

Following the earlier article on tensor products of vector spaces, we will now look at tensor products of modules over a ring R, not necessarily commutative. It turns out we have to distinguish between left and right modules now. Indeed recall … Continue reading →

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Tensor Product and Linear Algebra

Posted on January 29, 2015 by limsup

Tensor products can be rather intimidating for first-timers, so we’ll start with the simplest case: that of vector spaces over a field K. Suppose V and W are finite-dimensional vector spaces over K, with bases and respectively. Then the tensor product is the vector … Continue reading →

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Hom Functor

Posted on January 25, 2015 by limsup

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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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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