# 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.
• Day 1, 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.
• Day 2, Lecture 2: : Product and coproduct of modules, free modules.
• Day 3, Lecture 1: : Rings of fractions, extended and contracted ideals.
• Day 3, Lecture 2: : Exact sequences of modules, commutative diagrams, four lemma, five lemma.
Modules of fractions, local-global principles. Nakayama Lemma.
• Day 4, Lecture 1: : Basics of Krull dimension of a ring. Integral extensions, weak Nullstellensatz.
• Day 4, Lecture 2: : Dedekind domains, fractional ideals, class group.
• Day 5, Lecture 1: : Noetherian rings and modules, Hilbert basis theorem.
• Day 5, Lecture 2: : Modules over PIDs, Structure Theorem
• Day 6, Lecture 1: : Affine algebras over a field, Hilbert Nullstellensatz, basics of affine algebraic varieties.
• Day 6, 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 Algebra-II, Second edition. W. H. Freeman and Company, New York, 1989.

SECOND WEEK: Non-Commutative Algebra

• Day 1, Lecture 1: Basic differences between Commutative and Non-commutative rings, Examples of Non-commutative rings, One-sided ideals, Division rings, Simple rings, Ideals in matrix rings, Algebras.
• Day 1, 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 left-right symmetry, Modules of 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.
• Day 2, Lecture 2: Jacobson Radical, J-semisimple rings and their relationship with semisimple rings.
• Day 3, Lecture 1: Jacobson’s Density Theorem, Schur’s Lemma, Wedderburn-Artin Theorem for Rings and Algebras.
• Day 3, Lecture 2: Tensor Products of Modules and Algebras.
• Day 4, Lecture 1: Bimodules, Finite dimensional algebras over fields, Central Simple Algebras.
• Day 4, Lecture 2: Hom, Tensor Functors and their exactness.
• Day 5, Lecture 1: Skolem-Noether Theorem, Double Centraliser Theorem, Splitting Field of Central Simple Algebras.
• Day 5, 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.
• Day 6, Lecture 2: Von-Neumann 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. Springer-
Verlag, New York, 2001.
[2] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, 189. Springer-Verlag, New
York, 1999.

Speakers
1.Mrinal Kanti Das, ISI Kolkata
2.Utsav Choudhury, ISI Kolkata
3.Vikas Bist, 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 11.30-1.00 Lecture 2 2.30-3.30 Tutorial 1 4.00-5.00 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 VB DK VB, DK, TS VB, DK, TS July 2 VB DK DK, VB, TS DK, VB, TS July 3 VB DK AKB, DK, TS VB, DK, TS July 4 VB DK DK, VB, TS DK, VB, TS July 5 VB DK AKB, DK, TS VB, DK, TS July 6 VB DK DK, VB, TS DK, VB, TS

Tea Breaks: 11.00 - 11.30 and 3.30 - 4.00
Lunch Break: 1.00 - 2.30
Snacks: 5.00 - 5.30

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