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Tag Archives: prime ideals
Commutative Algebra 23
Localization and Spectrum Recall that the ideals of correspond to a subset of the ideals of A. If we restrict ourselves to prime ideals, we get the following nice bijection. Theorem 1. The above gives a bijection between Useful trick If … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, local rings, localization, prime ideals, rational functions, spectrum, zariski topology
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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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Topics in Commutative Rings: Unique Factorisation (3)
Example 1: The Gaussian Integers Z[i] Let’s pick the norm function N : Z[i]-{0} → N where N(a+bi) = (a+bi)(a–bi) = a2+b2. We know that N is a multiplicative function, i.e. N(r)N(s) = N(rs). Instead of checking this by brute force, we write N(x) = x·xc, where (a+bi)c = a-bi is the conjugate of a+bi. It’s easy to … Continue reading
Topics in Commutative Rings: Unique Factorisation (2)
In the previous article, we imposed certain finiteness conditions on the ring (specifically a.c.c. on principal ideals: that every increasing sequence of principal ideals is eventually constant), then proved that unique factorisation holds if and only if all irreducible elements … Continue reading
Posted in Notes
Tagged commutative rings, euclidean domains, irreducibles, prime ideals, primes, principal ideal domains, ring theory, rings, UFDs, unique factorisation
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Topics in Commutative Rings: Unique Factorisation (1)
Unique Factorisation: Basics Throughout this post, let R be an integral domain; recall that this means R is a commutative ring such that whenever ab=0, either a=0 or b=0. The simplest example of an integral domain is Z, the ring of integers. What’s of interest to … Continue reading
Posted in Notes
Tagged commutative rings, irreducibles, prime ideals, primes, ring theory, rings, UFDs, unique factorisation
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Introduction to Ring Theory (5)
Our first order of the day is to state the correspondence between the ideals and subrings of R/I and those of R. This is totally analogous to the case of groups. Theorem. Let I be an ideal of R. There are 1-1 … Continue reading
Posted in Notes
Tagged advanced, chinese remainder theorem, ideals, maximal ideals, prime ideals, ring theory, rings
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Mathematics and Such 