Section
Algebra: Commutative Algebra
Localization, Noetherian rings, primary decomposition, and integral extensions.
Core idea
This section covers commutative algebra: localization of rings and modules, local rings, Noetherian and Artinian rings, Krull dimension, integral extensions, and primary decomposition.
Definitions
Localization
- Localization of a ring
- Multiplicative set
- Localization at a prime ideal
- Local ring
- Maximal ideal of a local ring
- Residue field
- Localization of a module
- Extension of scalars
- Restriction of scalars
Spectrum and Dimension
- Prime spectrum (Spec R)
- Maximal spectrum (MaxSpec R)
- Zariski topology
- Krull dimension
- Height of a prime ideal
Integral Extensions
Primary Decomposition and Chain Conditions
- Primary decomposition (ideal-theoretic)
- Noetherian ring (ACC on ideals)
- Artinian ring (DCC on ideals)
Special Rings
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Theorems
- Correspondence of primes under localization
- Krull's principal ideal theorem
- Lasker–Noether primary decomposition theorem
- Going-up theorem
- Lying-over theorem
- Going-down theorem
- Nullstellensatz (ideal–variety correspondence)
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Lemmas
- Localization inverts precisely the multiplicative set
- Nakayama lemma
- Prime avoidance lemma
- Noether normalization lemma
- Jacobson radical = intersection of maximal ideals
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Propositions
- Localization is exact (flatness)
- Localization of Noetherian is Noetherian
- Localization preserves primality/maximality
- Jacobson radical annihilates simple modules
- Simple Artinian = matrix ring over division ring
- Semisimple Artinian = product of matrix rings
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Corollaries
- Every ideal in Noetherian ring has primary decomposition
- Hilbert basis theorem corollary (k[x₁,...,x_n] Noetherian)
- Nullstellensatz corollary (radical ideals ↔ affine algebraic sets)
- Localization corollary (localizations of Noetherian are Noetherian)
- Nakayama corollary (M f.g., IM = M, I ⊆ Jac(R) ⇒ M = 0)