Author Archives: limsup

Commutative Algebra 17

Posted on April 4, 2020 by limsup

Field of Fractions Throughout this article, A denotes an integral domain (which may not be a UFD). Definition. The field of fractions of A is an embedding of A into a field K, such that every element of K can be … Continue reading →

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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 15

Posted on April 2, 2020 by limsup

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 →

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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 12

Posted on March 30, 2020 by limsup

Some Results on Posets In this article we have two goals in mind: to introduce the idea of noetherian posets, and to state Zorn’s lemma and give some examples. The latter is of utmost importance in diverse areas of mathematics. … Continue reading →

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Commutative Algebra 11

Posted on March 29, 2020 by limsup

Coordinate Rings as k-algebras Let k be an algebraically closed field. Recall that a closed subset is identified by its coordinate ring k[V], which is a finitely generated k-algebra since Definition. An affine k-variety is a finitely generated k-algebra A which is … 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 →

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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 →

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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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