Tag Archives: rings

Commutative Algebra 13

Zariski Topology for Rings In this article, we generalize earlier results in algebraic geometry to apply to general rings. Recall that points on an affine variety V correspond to maximal ideals . For general rings, we have to switch to … Continue reading

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Commutative Algebra 10

Algebras Over a Ring Let A be any ring; we would like to look at A-modules with a compatible ring structure. Definition. An –algebra is an -module , together with a multiplication operator such that becomes a commutative ring (with 1); multiplication … Continue reading

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Commutative Algebra 9

Direct Sums and Direct Products Recall that for a ring A, a sequence of A-modules gives the A-module where the operations are defined component-wise. In this article, we will generalize the construction to an infinite collection of modules. Throughout this article, let denote … Continue reading

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Commutative Algebra 8

Generated Submodule Since the intersection of an arbitrary family of submodules of M is a submodule, we have the concept of a submodule generated by a subset. Definition. Given any subset , let denote the set of all submodules of M containing … Continue reading

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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

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Commutative Algebra 6

Injective and Surjective Maps Proposition. 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 W. … Continue reading

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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

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