IST  Rings and Modules (2019)
Speakers and Syllabus
FIRST WEEK:Commutative Algebra
 Day 1
Lecture 1: Commutative rings with unity, morphism of rings, ideals. Existence of maximal ideals. Prime ideals, nil radical. Polynomial rings.
Lecture 2: Modules over commutative rings, morphism of modules, submodules, quotient modules, homomorphism theorems.  Day 2
Lecture 1: : Chinese remainder theorem, Division algorithm, Euclidean Domain, Principal Ideal Domain and Unique Factorization Domain.
Lecture 2: : Product and coproduct of modules, free modules.  Day 3
Lecture 1: : Rings of fractions, extended and contracted ideals.
Lecture 2: : Exact sequences of modules, commutative diagrams, four lemma, five lemma. Modules of fractions, localglobal principles. Nakayama Lemma.  Day 4
Lecture 1: : Basics of Krull dimension of a ring. Integral extensions, weak Nullstellensatz.
Lecture 2: : Dedekind domains, fractional ideals, class group.  Day 5
Lecture 1: : Noetherian rings and modules, Hilbert basis theorem.
Lecture 2: : Modules over PIDs, Structure Theorem  Day 6
Lecture 1: : Affine algebras over a field, Hilbert Nullstellensatz, basics of affine algebraic varieties.
Lecture 2: Lecture 2: Application of theory of finitely generated modules over PIDs to Linear Algebra.
Text Books:
[1] M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Levant Books, Kolkata, 2007.
[2] N. Jacobson, Basic AlgebraII, Second edition. W. H. Freeman and Company, New York, 1989.
SECOND WEEK: NonCommutative Algebra
 Day 1
Lecture 1: Basic differences between Commutative and Noncommutative rings, Examples of Noncommutative rings, Onesided ideals, Division rings, Simple rings, Ideals in matrix rings, Algebras.
Lecture 2: Modules, Basic differences between vector spaces and modules, Chain conditions for modules and rings, One sided Artinian and Noetherian rings and failure of leftright symmetry, Modulesof finite length, Composition series, Jordan Holder Theorem (statement only).  Day 2
Lecture 1: Semisimple Rings, Pierce Decompositions, Semisimple rings as direct products, Simple Artinian Rings.
Lecture 2: Jacobson Radical, Jsemisimple rings and their relationship with semisimple rings.  Day 3
Lecture 1: Jacobson’s Density Theorem, Schur’s Lemma, WedderburnArtin Theorem for Rings and Algebras.
Lecture 2: Tensor Products of Modules and Algebras.  Day 4
Lecture 1: Bimodules, Finite dimensional algebras over fields, Central Simple Algebras.
Lecture 2: Hom, Tensor Functors and their exactness.  Day 5
Lecture 1: SkolemNoether Theorem, Double Centraliser Theorem, Splitting Field of Central Simple Algebras.
Lecture 2: Projective, Injective and Flat Modules.  Day 6
Lecture 1: Wedderburn Theorem for finite division rings, Frobenius Theorem for division algebras over reals, Brauer Groups of fields.
Lecture 2: VonNeumann Rings as generalization of semisimple rings and their characterization as rings over which every module is flat.
Text Books:
[1] T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131. SpringerVerlag, New York, 2001.
[2] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, 189. SpringerVerlag, New York, 1999.
Speakers
1.Mrinal Kanti Das, ISI Kolkata
2.Utsav Choudhury, ISI Kolkata
3.A. K. Bhandari, Panjab University, Chandigarh
4.Dinesh Khurana, Panjab University, Chandigarh
Course Associates
1. Ardeline M. Buhphang, NEHU Shillong
2. Tikaram Subedi, NIT. Meghalaya.
Time Table
9.30  11.00 Lecture 1 
11.301.00 Lecture 2 
2.303.30 Tutorial 1 
4.005.00 Tutorial 2 

June 24  UC  MKD  UC, MKD, AMB  UC, MKD, AMB 
June 25  UC  MKD  MKD, UC, AMB  MKD, UC, AMB 
June 26  UC  MKD  UC, MKD, AMB  UC, MKD, AMB 
June 27  UC  MKD  MKD, UC, AMB  MKD, UC, AMB 
June 28  UC  MKD  UC, MKD, AMB  UC, MKD, AMB 
June 29  UC  MKD  MKD, UC, AMB  MKD, UC, AMB 
July 1  AKB  DK  AKB, DK, TS  AKB, DK, TS 
July 2  AKB  DK  DK, AKB, TS  DK, AKB, TS 
July 3  AKB  DK  AKB, DK, TS  AKB, DK, TS 
July 4  AKB  DK  DK, AKB, TS  DK, AKB, TS 
July 5  AKB  DK  AKB, DK, TS  AKB, DK, TS 
July 6  AKB  DK  DK, AKB, TS  DK, AKB, TS 
Tea Breaks: 11.00  11.30 and 3.30  4.00
Lunch Break: 1.00  2.30
Snacks: 5.00  5.30