Syllabus for Commutative Algebra I, M Math

1. Rings and ideals: review of ideals in quotient rings; prime and maximal ideals, prime ideals under quotient, existence of maximal ideals; operations on ideals (sum, product, quotient and radical); Chinese Remainder theorem; nilradical and Jacobson radical; extension and contraction of ideals under ring homomorphisms; prime avoidance.
   
   
2. Free modules; Projective Modules; Tensor Product of Modules and Algebras; Flat, Faithfully Flat and Finitely Presented Modules; Shanuels Lemma.
   
   
3. Localisation and local rings, universal property of localisation, extended and contracted ideals and prime ideals under localisation, localisation and quotients, exacteness property. Results on prime ideals like theorems of Cohen and Isaac. Nagatas criterion for UFD and applications; equivalence of PID and one-dimensional UFD.
   
   
4. Modules over local rings. Cayley-Hamilton, NAK lemma and applications. Examples of local-global principles. Projective and locally free modules. Patching up of Localisation.
   
   
5. Polynomial and Power Series Rings. Noetherian Rings and Modules. Hilberts Basis Theorem. Associated Primes and Primary Decomposition. Artininan Modules. Modules of Finite Length.
   
   
6. Integral Extensions: integral closure, normalisation and normal rings. Cohen-Seidenberg Going-Up Theorem. Hilberts Nullstellensatz and applications.
   
   
7. Valuations, Discrete Valuation Rings, Dedekind domains.