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Author Archives: limsup
Commutative Algebra 44
Fractional Ideals Throughout this article, let A be an integral domain and K its field of fractions. We do not assume the ring to be noetherian. The objective here is to develop the theory of multiplying and dividing certain classes of nonzero … Continue reading
Posted in Advanced Algebra
Tagged fractional ideals, ideal division, invertible ideals, localization, picard group, principal ideals
5 Comments
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
2 Comments
Commutative Algebra 42
Noether Normalization Theorem Throughout this article, k is a field, not necessarily algebraically closed. Definition. Let A be a finitely generated kalgebra which is an integral domain. We say are algebraically independent over k if they are so as elements … Continue reading
Commutative Algebra 41
Basic Definitions The objective of this article is to establish the theory of transcendence bases for field extensions. Readers who are already familiar with this may skip the article. We will focus on field extensions here. Definition. Let be a … Continue reading
Posted in Advanced Algebra
Tagged algebraic extensions, fields, transcendence basis, transcendence degree
2 Comments
Commutative Algebra 40
More on Integrality Lemma 1. Let be an integral extension. If is an ideal and , the resulting injection is an integral extension. Proof Any element of can be written as , . Then x satisfies a monic polynomial relation: . … Continue reading
Posted in Advanced Algebra
Tagged closed maps, fibres, finite extensions, going up, integral extensions, krull dimension, localization
4 Comments
Commutative Algebra 39
Integrality Throughout this article, A is a subring of B; we will also call B a ring extension of A. Definition. An element is said to be integral over A if we can find (where ) such that in B. For example, is integral over since … Continue reading
Posted in Advanced Algebra
Tagged field of fractions, finite extensions, integral closure, integral extensions, normal domains, rings, UFDs
4 Comments
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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