Category Archives: Advanced Algebra

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 12

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

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

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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