- #31
elias001
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@fresh_42 your post #6 points 2.)-4.) is where you try to shown that statement?
##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?
- Context: Undergrad
- Thread starter elias001
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- Abstract algebra
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SUMMARY
The discussion centers on the isomorphism between the localization of a ring at a prime ideal and the quotient of the localization of the ring by an ideal. Specifically, Proposition 2.1.1 establishes that for an ideal \(\mathfrak{a}\) of a ring \(A\), the localization \(A_\mathfrak{p}/\mathfrak{a}_\mathfrak{p}\) is isomorphic to \((A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}\). The proof utilizes the First Isomorphism Theorem, demonstrating that the map \(f: A_\mathfrak{p} \to (A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}\) defined by \(\frac{m}{s} + \mathfrak{a}_\mathfrak{p} \mapsto \frac{m + \mathfrak{a}}{s + \mathfrak{a}}\) is surjective with kernel \(\mathfrak{a}A_\mathfrak{p}\). The discussion also addresses the bijection between prime ideals of \(A\) and \(A_\mathfrak{p}\) through the extension of ideals.
PREREQUISITES- Understanding of localization in ring theory, specifically \(A_\mathfrak{p}\).
- Familiarity with ideals and quotient rings, particularly \(\mathfrak{a}_\mathfrak{p}\).
- Knowledge of the First Isomorphism Theorem in algebra.
- Basic concepts of prime ideals and their extensions in commutative algebra.
- Study the First Isomorphism Theorem in detail to understand its applications in ring theory.
- Explore the properties of localization in commutative algebra, focusing on \(A_\mathfrak{p}\) and its maximal ideal.
- Investigate the relationship between prime ideals in a ring and their extensions in localized rings.
- Examine examples of localization and quotient rings to solidify understanding of the
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