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Tag Archives: rings
Commutative Algebra 39
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
Posted in Advanced Algebra
Tagged field of fractions, finite extensions, integral closure, integral extensions, normal domains, rings, UFDs
4 Comments
Commutative Algebra 18
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 algebra, category theory, contravariant, coslice categories, covariant, functors, morphisms, rings
2 Comments
Commutative Algebra 16
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
Commutative Algebra 14
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
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 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
5 Comments
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
5 Comments
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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Commutative Algebra 6
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 algebraic geometry, ideals, irreducible spaces, monomials, rings
4 Comments