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 finitely-generated. Let K(R) be the Grothendieck group of all finitely-generated R-modules; K(R) is the free abelian group generated by [M] for simple … Continue reading

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Exact Sequences and the Grothendieck Group

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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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

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