Tag Archives: rings

Commutative Algebra 39

Posted on April 27, 2020 by limsup

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 →

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Commutative Algebra 18

Posted on April 5, 2020 by limsup

Basics of Category Theory As we proceed, we should cover some rudimentary category theory or many of the subsequent constructions would seem unmotivated. The essence of category is in studying algebraic objects and the homomorphisms between them. By now we have … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , , , | 2 Comments

Commutative Algebra 16

Posted on April 3, 2020 by limsup

Gcd and Lcm We assume A is an integral domain throughout this article. If A is a UFD, we can define the gcd (greatest common divisor) and lcm (lowest common multiple) of two elements as follows. For , we can write the … Continue reading →

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Commutative Algebra 14

Posted on April 1, 2020 by limsup

Basic Open Sets For , let , an open subset of Spec A. Note that . Proposition 1. The collection of over all forms a basis for the topology of . Proof Let be an open subset of Spec A. Suppose … Continue reading →

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Commutative Algebra 13

Posted on March 31, 2020 by limsup

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

Posted on March 28, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged , , , , , , | 5 Comments

Commutative Algebra 9

Posted on March 27, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged , , , , | 5 Comments

Commutative Algebra 8

Posted on March 26, 2020 by limsup

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

Posted on March 25, 2020 by limsup

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

Posted on March 24, 2020 by limsup

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 , , , , | 4 Comments