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Tag Archives: noetherian
Commutative Algebra 55
Exactness of Completion Throughout this article, A denotes a filtered ring. Proposition 1. Let be a short exact sequence of Amodules. Suppose M is filtered, inducing filtrations on N and P. Then is also exact as modules. Proof Without loss of … Continue reading
Posted in Advanced Algebra
Tagged aadic filtrations, artinrees lemma, blowup algebras, completions, filtrations, limits, noetherian
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Commutative Algebra 43
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
Posted in Advanced Algebra
Tagged algebras, catenary rings, krull dimension, noether normalization, noetherian
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Commutative Algebra 38
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
Commutative Algebra 37
Artinian Modules Instead of the ascending chain condition, we can take its reverse. Definition. Let M be an Amodule. Consider the set of submodules of M, ordered by inclusion, i.e. if and only if . We say M is artinian … Continue reading
Posted in Advanced Algebra
Tagged artinian, composition factors, composition series, length of module, noetherian, simple modules
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Commutative Algebra 36
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 nonempty space X satisfying any of the … Continue reading
Commutative Algebra 35
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
Posted in Advanced Algebra
Tagged flat modules, hilbert basis theorem, ideals, modules, noetherian, projective modules
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Commutative Algebra 15
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
Posted in Advanced Algebra
Tagged integral domains, irreducibles, noetherian, primes, UFDs, unique factorisation
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