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Category Archives: Notes
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
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Tagged bimodules, hom functor, leftexact, modules, rightexact, tensor products
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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
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Tagged bilinear maps, duals, linear algebra, tensor algebra, tensor products, universal properties
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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
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Tagged bimodules, hom functor, left modules, leftexact, modules, right modules
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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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