Course Archives Theoretical Statistics and Mathematics Unit
Course: Commutative Algebra I
Time: Currently not offered
Syllabus
Past Exams

Syllabus:

i) Quick review of Rings and ideals: ideals in quotient rings; prime ideals under quotient, existence of maximal ideals; operations on ideals (sum, product, quo tient and radical); ideals and prime ideals in product rings, Chinese Remainder theorem; nilradical and Jacobson radical; extension and contraction of ideals under ring homomorphisms;
ii) Free modules; Projective Modules; Tensor Product of Modules and Algebras; Flat, Faithfully Flat and Finitely Presented Modules; Shanuels Lemma.
iii) Localisation and local rings, universal property of localisation, extended and contracted ideals and prime ideals under localisation, localisation and quotients, exacteness property, Nagatas criterion for UFD and applications.
iv) Prime avoidance. Results on prime ideals like theorems of Cohen and Isaac, equivalence of PID and one-dimensional UFD.
v) Modules over local rings. Cayley-Hamilton, NAK lemma and applications. Examples of local-global principles. Projective and locally free modules. Patching up of Localisation.
vi) Polynomial and Power Series Rings. Noetherian Rings and Modules. Hilberts Basis Theorem, Graded Rings, equivalence of Noetherian rings and finitely generated algebras for graded rings.
vii) Integral Extensions: integral closure, normalisation and normal rings. Cohen Seidenberg Going-Up Theorem. Hilberts Nullstellensatz and applications. In troduction to Valuation rings.
viii) Time permitting: Introduction to Grobner basis.

Suggested Texts :
(a) N.S. Gopalakrishnan, Commutative Algebra, Oxonian Press (1984).
(b) M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addision Wesley(1969).
(c) M. Reid: Undergraduate commutative algebra, LMS Student Texts (29), Cam bridge Univ. Press (1995).
(d) R.Y. Sharp: Steps in commutative algebra, LMS Student Texts, Cambridge Univ. Press.
(e) E. Kunz: Introduction to commutative algebra and algebraic geometry, Birkhauser (1985).
(f) D.S. Dummit and R.M. Foote: Abstract Algebra (Part V), John Wiley (2003).
(g) D. Eisenbud: Commutative algebra with a view toward algebraic geometry GTM (150), Springer-Verlag (1995).

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Past Exams
Midterm Semestral Supplementary and Back Paper
 23.pdf

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