Tag Archives: noetherian

Commutative Algebra 55

Posted on May 13, 2020 by limsup

Exactness of Completion Throughout this article, A denotes a filtered ring. Proposition 1. Let be a short exact sequence of A-modules. Suppose M is filtered, inducing filtrations on N and P. Then is also exact as -modules. Proof Without loss of … Continue reading →

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Commutative Algebra 43

Posted on May 1, 2020 by limsup

Catenary Rings Let us look at prime chains in greater detail. Definition. Let be a chain of prime ideals of a ring A. We say the chain is saturated if for any prime ideal of A, ; maximal if it … Continue reading →

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Commutative Algebra 38

Posted on April 26, 2020 by limsup

Artinian Rings The main result we wish to prove is the following. Theorem. A ring A is artinian if and only if it is noetherian and , where denotes the Krull dimension. Note Recall that means all prime ideals of A … 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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Commutative Algebra 36

Posted on April 24, 2020 by limsup

In this article, we will study the topology of Spec A when A is noetherian. For starters, let us consider irreducible topological spaces in greater detail. Irreducible Spaces Recall that an irreducible topological space is a non-empty space X satisfying any of the … Continue reading →

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Commutative Algebra 35

Posted on April 23, 2020 by limsup

Noetherian Modules Through this article, A is a fixed ring. For the first two sections, all modules are over A. Recall that a submodule of a finitely generated module is not finitely generated in general. This will not happen if we constrain … Continue reading →

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Commutative Algebra 15

Posted on April 2, 2020 by limsup

Unique Factorization Through this article and the next few ones, we will explore unique factorization in rings. The inspiration, of course, comes from ℤ. Here is an application of unique factorization. Warning: not all steps may make sense to the … Continue reading →

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Commutative Algebra 12

Posted on March 30, 2020 by limsup

Some Results on Posets In this article we have two goals in mind: to introduce the idea of noetherian posets, and to state Zorn’s lemma and give some examples. The latter is of utmost importance in diverse areas of mathematics. … Continue reading →

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Jacabson Radical

Posted on January 13, 2015 by limsup

Recall that the radical of the base ring R is called its Jacobson radical and denoted by J(R); this is a two-sided 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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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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