Mathematics / advanced
Commutative algebra
Analyze rings locally and globally using modules, localization, dimension, integrality, and homological invariants.
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Inside this module
Module 01 advanced
Commutative algebra
Analyze rings locally and globally using modules, localization, dimension, integrality, and homological invariants.
- 01
Modules, submodules and exact sequences
Replace vector-space intuition with ring-linear structure and record generators, relations and missing image information precisely.
29 min - 02
Noetherian rings and Hilbert’s basis theorem
Turn ascending-chain control into finite generators and prove why adjoining finitely many polynomial variables preserves it.
31 min - 03
Localization and local rings
Invert a controlled set of scalars, track which elements disappear and isolate the algebra near a prime ideal.
30 min - 04
Prime spectra and the Zariski topology
Treat primes as points, calculate algebraic closed sets and see why topology alone cannot retain nilpotent structure.
29 min - 05
Integral extensions and going up
Use monic equations to control algebraic extensions and distinguish the direction and hypotheses of prime-chain lifting theorems.
32 min - 06
Primary decomposition and associated primes
Separate component support from embedded annihilator information and calculate an irredundant decomposition with a genuine embedded prime.
33 min - 07
Krull dimension and height
Measure prime-chain length, distinguish local height from quotient dimension and apply generator bounds only to minimal primes.
32 min - 08
Regular sequences and depth
Measure successive nonzerodivisor cuts in a finite module over a Noetherian local ring and distinguish maximal sequence length from an arbitrary first choice.
34 min - 09
Completions and Nakayama’s lemma
Control compatible finite-order data and lift local generators with the exact finiteness and Jacobson-radical assumptions exposed.
34 min - 10
Flatness and base change
Separate preservation of exactness from faithful detection and track what changes when scalars are extended or specialized.
34 min