Local rings,” Section 10.18¶
Authors, T. S. P. Local rings,” Section 10.18.
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Domain-specific¶
- Maximal Ideal
- … extensions: if \(f\in k[x]\) is irreducible, then \((f)\) is maximal and \(k[x]/(f)\) is a field. In commutative algebra, localization at a prime \(\mathfrak p\) produces a local ring \(R_{\mathfrak p}\) whose unique maximal ideal is \(\mathfrak pR_{\mathfrak p}\) and whose residue field is \(\kappa(\mathfrak p)\).
This sourceDefines local rings and residue fields and proves \(R_{\mathfrak p}\) is local with maximal ideal \(\mathfrak pR_{\mathfrak p}\).
- … extensions: if \(f\in k[x]\) is irreducible, then \((f)\) is maximal and \(k[x]/(f)\) is a field. In commutative algebra, localization at a prime \(\mathfrak p\) produces a local ring \(R_{\mathfrak p}\) whose unique maximal ideal is \(\mathfrak pR_{\mathfrak p}\) and whose residue field is \(\kappa(\mathfrak p)\).
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