
Recent Posts
Archives
 June 2018
 July 2016
 June 2016
 May 2016
 March 2015
 February 2015
 January 2015
 December 2014
 December 2013
 November 2013
 July 2013
 June 2013
 May 2013
 March 2013
 February 2013
 January 2013
 December 2012
 November 2012
 October 2012
 September 2012
 August 2012
 April 2012
 March 2012
 February 2012
 January 2012
 December 2011
 November 2011
 October 2011
Categories
Meta
Pages
Monthly Archives: January 2015
Tensor Product over Noncommutative Rings
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
Posted in Notes
Tagged bimodules, hom functor, leftexact, modules, rightexact, tensor products
Leave a comment
Tensor Product and Linear Algebra
Tensor products can be rather intimidating for firsttimers, so we’ll start with the simplest case: that of vector spaces over a field K. Suppose V and W are finitedimensional vector spaces over K, with bases and respectively. Then the tensor product is the vector … Continue reading
Posted in Notes
Tagged bilinear maps, duals, linear algebra, tensor algebra, tensor products, universal properties
Leave a comment
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
Posted in Notes
Tagged bimodules, hom functor, left modules, leftexact, modules, right modules
Leave a comment
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
Posted in Notes
Tagged composition series, exact sequences, grothendieck group, modules, short exact sequences
Leave a comment
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
Posted in Notes
Tagged indecomposable modules, krullschmidt, local rings, matrix rings, splitting lemma, unique factorisation
Leave a comment
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
Leave a comment
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
Posted in Notes
Tagged artinian, hopkinslevitzki, jacobson radical, matrix rings, nilpotent ideals, noetherian, semisimple rings
Leave a comment