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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
Posted in Advanced Algebra
Tagged homomorphism, maximal ideals, prime ideals, rings, spectrum, topology, zariski topology
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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
Posted in Advanced Algebra
Tagged axiom of choice, noetherian, posets, totally ordered sets, zorn's lemma
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Commutative Algebra 11
Coordinate Rings as kalgebras 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 kalgebra since Definition. An affine kvariety is a finitely generated kalgebra A which is … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, algebras, cotangent spaces, maximal ideals, tangent spaces, varieties
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Commutative Algebra 10
Algebras Over a Ring Let A be any ring; we would like to look at Amodules 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 algebras, generated submodules, homomorphism, modules, quotient modules, rings, submodules
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Commutative Algebra 9
Direct Sums and Direct Products Recall that for a ring A, a sequence of Amodules gives the Amodule where the operations are defined componentwise. 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 direct products, direct sums, modules, rings, universal properties
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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
Posted in Advanced Algebra
Tagged free modules, generated submodules, homomorphism, modules, quotient modules, rings, submodules
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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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