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Tag Archives: ideals
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 22
Localization Recall that given an integral domain, there is a canonical way to construct the “smallest field containing it”, its field of fractions. Here, we will generalize this construction to arbitrary rings. We let A be a fixed ring throughout. Definition. … Continue reading
Posted in Advanced Algebra
Tagged field of fractions, ideals, localization, universal properties
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Commutative Algebra 7
Modules Having dipped our toes into algebraic geometry, we are back in commutative algebra. Next we would like to introduce “linear algebra” over a ring A. Most of the proofs should pose no difficulty to the reader so we will … Continue reading
Posted in Advanced Algebra
Tagged ideals, linear algebra, module homomorphism, modules, rings, submodules
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Commutative Algebra 6
Injective and Surjective Maps Proposition 1. Let be a morphism of closed sets, with corresponding . is injective if and only if is dense. is surjective if and only if is an embedding of V as a closed subspace of … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, ideals, irreducible spaces, monomials, rings
4 Comments
Commutative Algebra 5
Morphisms in Algebraic Geometry Next we study the “nice” functions between closed subspaces of . Definition. Suppose and are closed subsets. A morphism is a function which can be expressed as: for some polynomials . We also say f is a regular … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, continuity, ideals, morphisms, rings, topology
2 Comments
Commutative Algebra 4
More Concepts in Algebraic Geometry As before, k denotes an algebraically closed field. Recall that we have a bijection between radical ideals of and closed subsets of . The bijection reverses the inclusion so if and only if . Not too … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, closed subsets, ideals, irreducible spaces, rings, zariski topology
7 Comments
Commutative Algebra 3
Algebraic Geometry Concepts We have decided to introduce, at this early point, some basics of algebraic geometry in order to motivate the later concepts. In summary, algebraic geometry studies solutions to polynomial equations over a field. First we consider a … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, commutative rings, ideals, nullstellensatz, radical ideals, rings, topology, zariski topology
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Commutative Algebra 2
Radical of an Ideal In this installation, we will study more on ideals of a ring A. Definition. If is an ideal, its radical is defined by To fix ideas, again consider the case again. For the ideal (m) where , … Continue reading
Posted in Advanced Algebra
Tagged commutative rings, ideal division, ideals, radical ideals, rings
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Commutative Algebra 1
More About Ideals Recall that we defined three operations on ideals: intersection, sum and product. We can take intersection and sum of any collection of ideals (even infinitely many of them), but we can only define the product of finitely many … Continue reading
Posted in Advanced Algebra
Tagged chinese remainder theorem, coprime ideals, ideals, rings
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Commutative Algebra 0
We’re starting a new series on commutative algebra. This has been in the works for way too long, and eventually we just decided to push ahead with it anyway. Most of the articles will be short, and we’ll try to … Continue reading
Posted in Advanced Algebra
Tagged commutative rings, fields, ideals, integral domains, rings
2 Comments