**[Cr:4, Lc:3, Tt:1, Lb:0]**

- Recapitulation: Ideals, factorization rings, prime and maximal ideals, modules.
- Nilradical and Jacobson radical, extensions and contractions of ideals.
- Localization of rings and modules.
- Integral dependence, integrally closed domains, going up and going down theorem, valuation rings.
- Noetherian and Artinian rings, chain conditions on modules.
- Exact sequences of modules, tensor product, projective and injective modules.
- Basics of categories and functors.
- Exact sequences and complexes in categories, additive functors, derived functors EXT and TOR functors.
- Discrete valuation rings and Dedekind domains.

- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison Wesley (1969).
- Nathan Jacobson, Basic Algebra Vol. II, Dover Publications (2009).
- Joseph J. Rotman, An Introduction to Homological Algebra, ( 2nd edition), Springer (2008).
- L. R. Vermani, An Elementary Approach to Homological Algebra, Chapman & Hall/CRC (2003).