Contents
Unit I - Rings and Ideals:
Zero divisors, Nilpotent elements, Units, Prime ideals and maximal ideals, Nilradical and Jacobson
radical, Operations on ideals, Extensions and contraction of ideals. (18 Hours)
Unit II - Modules:
Recapitulation of Operations on submodules, Isomorphism theorems. Direct sum and product,
Finitely generated modules, Nakayama’s lemma, Exact sequences (omit tensor products and related
results). (12 Hours)
Unit III –Modules of Fractions and Primary Decomposition,:
Local properties, Extended and contracted ideals in rings of fractions, First and second uniqueness
theorems. (12 Hours)
Unit IV - Integral Dependence and Chain Conditions:
Integral dependence, The going-up theorem, Integrally closed integral domains, The going-down
theorem, Noetherian rings and modules, Primaryde composition in Noetherian rings.
(6 Hours)
References
[1] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Indian Ed., Lavant
Books, 2007.
[2] N. Bourbaki, Commutative Algebra, American Mathematical Society, 1972.
[3] N. S. Gopalkrishnan, Commutative Algebra, 2nd Ed., University Press, 2015.
- Teacher: Dr Chandru Hegde