Course Archives: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore

Course Archives Theoretical Statistics and Mathematics Unit

Course: Rings and Modules
Instructor: Sury B
Room: Second floor auditorium
Level: Undergraduate
Time: Currently offered

Syllabus
Past Exams

Syllabus:

i) Rings, Left and Right ideals, Examples of Polynomial rings, Matrix rings and Group rings, Quotient rings by two-sided ideals.
ii) Commutative rings: Units, Nilpotents, Adjunction of elements, Chinese remainder theorem, Maximal and prime ideals, Localization.
iii) Factorisation theory in domains: Irreducible and prime elements, Euclidean domains, Principal Ideal Domains, Unique Factorisation Domains, Gausss lemma, Eisensteins Criterion.
iv) Noetherian rings, Hilbert basis theorem.
v) Modules: Structure of finitely generated modules over a PID and their representation matrices, Applications to Rational canonical form and Jordan form of a matrix.

Reference Texts:

(a) M. Artin: Algebra.
(b) S. D. Dummit and M. R. Foote: Abstract Algebra.
(c) I. N. Herstein: Topics in Algebra.
(d) K. Hoffman and R. Kunze: Linear Algebra.
(e) C. Musili: Rings and Modules.
(f) J. A. Gallian: Contemporary Abstract Algebra.
(g) N. Jacobson: Basic Algebra.

Evaluation: