Tag Archives: composition series

Commutative Algebra 59

Posted on May 17, 2020 by limsup

Prime Composition Series Throughout this article, A is a noetherian ring and all A-modules are finitely generated. Recall (proposition 1 here) that if M is a noetherian and artinian module, we can find a sequence of submodules whose consecutive factors … Continue reading →

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Commutative Algebra 37

Posted on April 25, 2020 by limsup

Artinian Modules Instead of the ascending chain condition, we can take its reverse. Definition. Let M be an A-module. Consider the set of submodules of M, ordered by inclusion, i.e. if and only if . We say M is artinian … Continue reading →

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Projective Modules and the Grothendieck Group

Posted on February 5, 2015 by limsup

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

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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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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