Learn · Introduction to Commutative Algebra
Lecture 3 of 39Ring theory basics
Ideals in commutative rings
Prof. A. V. Jayanthan · IIT Madras
Credits and licence
- From
- Introduction to Commutative Algebra, IIT Madras (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- Tweens profiles and older. How courses are placed
All the lectures
- Review of Ring Theory
- Review of Ring Theory (Ctd.)
- Ideals in commutative rings
- Operations on ideals
- Properties of prime ideals
- Colon and Radical of ideals
- Radicals, extension and contraction of ideals
- Modules and homomorphisms
- Isomorphism theorems and Operations on modules
- Operations on modules (Ctd.)
- Module homomorphism and determinant trick
- Nakayama’s lemma and exact sequences
- Exact sequences (Ctd.)
- Homomorphisms and Tensor products
- Properties of tensor products
- Properties of tensor products (Ctd.)
- Tensor product of Algebras
- Localization
- Localization (Ctd.)
- Local properties
- Further properties of localization
- Intergral dependence
- Integral extensions
- Lying over and Going-up theorems
- Going-down theorem
- Going-down theorem (Ctd.)
- Chain conditions
- Noetherian and Artinian modules
- Properties of Noetherian and Artinian modules, Composition Series
- Further properties of Noetherian and Artinian modules and rings
- Hilbert basis theorem and Primary decomposition
- Primary decomposition (Ctd.)
- Uniqueness of primary decomposition
- 2nd Uniqueness theorem, Artinian rings
- Properties of Artinian rings
- Structure Theorem of Artinian rings
- Noether Normalization
- Hilberts Nullstellensatz
- Live Session 02-03-2021