Titlebook: Commutative Algebra through Exercises; Andrea Bandini,Patrizia Gianni,Enrico Sbarra Textbook 2024 The Editor(s) (if applicable) and The Au

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Affine Algebraic Varieties varieties. In addition, we explore the properties of these varieties and ideals by using the theory of ideals developed so far. Furthermore, we provide a proof of the Hilbert Nullstellensatz, which establishes a fundamental relation between geometry and algebra. We then apply this theorem, along wi

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Localizationring ., or, in greater generality, the construction of the field of fractions of a domain. We will define fractions using as denominators elements of a chosen subset of the ring. We will see that, when we choose the denominators in the complement of a prime ideal, this construction produces a local

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Noetherian and Artinian Rings. Primary Decomposition finitely generated and can be decomposed as a finite intersection of primary ideals. Additionally, we prove that . is Noetherian if and only if . is Noetherian, and that . is Artinian if and only if, up to isomorphism, it is a finite direct sum of local Artinian rings. This result generalizes what

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Localizationa chosen subset of the ring. We will see that, when we choose the denominators in the complement of a prime ideal, this construction produces a local ring whose maximal ideal is the extension of that prime with respect to a canonical homomorphism.