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Tag Archives: composition series
Projective Modules and the Grothendieck Group
This is a continuation of the previous article. Throughout this article, R is an artinian ring (and hence noetherian) and all modules are finitelygenerated. Let K(R) be the Grothendieck group of all finitelygenerated Rmodules; K(R) is the free abelian group generated by [M] for simple … Continue reading
Posted in Notes
Tagged artinian, composition series, grothendieck group, projective 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
Posted in Notes
Tagged composition series, exact sequences, grothendieck group, modules, short exact sequences
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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
Posted in Notes
Tagged algebra, artinian, composition series, length of module, matrix rings, modules, noetherian, unique factorisation
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