IST - Rings and Modules (2019)

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 Venue: North-Eastern Hill University,  Shillong, Meghalaya Date: 24th, Jun 2019 to 6th, Jul 2019

 School Convener(s) Name Dinesh Khurana Ardeline Buhphang Mailing Address Panjab University, Chandigarh NEHU Shillong

Speakers with their affiliations:

 Name of the speaker Affiliation Mrinal Kanti Das ISI Kolkata Utsav Choudhury ISI Kolkata Vikas Bist Panjab University, Chandigarh Dinesh Khurana Panjab University, Chandigarh

Course Associates

 Name Affiliation Ardeline M. Buhphang NEHU, Shillong Ms. Parnashree Ghosh ISI Kolkata

Syllabus:

 Name of the speaker Affiliation Detailed syllabus Mrinal Kanti Das ISI Kolkata Lecture 1, June 24. Spec(Z), Ideals of polynomial rings, Substitution Principle, Vector Fields, Derivations. Lecture 2, June 25. Local and Semi-local rings, Comaximality, Chinese Remainder Theorem, Factorization Theory in Domains, Euclidean Domains (ED) and Norm, Principal Ideals, Principal Ideal Domains (PID) and their Examples. Lecture 3, June 26. Difference between ED, PID and Uuique Factorization Domains, Construction of Field of Fractions and their properties, Construction and Properties of S −1 R for multiplicative S ⊆ R, Dedekind Domains. Lecture 4, June 27. Construction and Properties of S −1 M for multiplicative S ⊆ R and R-module M , Extension and Contraction of ideals, Local-Global Principle, Projective modules, Noetherian ring, Finitely Generated K-Algebra, Applications of Nakayama Lemma, Hilbert Basis Theorem, Ideal Class Groups. Lecture 5, June 28. Integral Elements and Integral Closures, Integrally Closed Rings and their Properties, Going-up and Going-down Theorems. Lecture 6, June 29. Krull Dimension and Height of a Prime Ideal. Utsav Choudhury ISI Kolkata Lecture 1, June 24. Motivation for Studying Modules, Group actions, Homomorphisms and Isomorphisms of modules, Quotient and submodules, Correspondence theorem. Lecture 2, June 25. Construction of New Modules from Given Modules, Direct Product and Direct Sums of Modules and Universal Property, Finitely generated modules, Free modules. Lecture 3, June 26. Nakayama Lemma, Multiplicative Closed sets in R, Rings and Modules of fractions, Local-Global principle, Exact Sequences, Sketches of Proofs for Four Lemma and Five Lemma using diagram chasing techniques. Lecture 4, June 27. Modules over PIDs and its applications, Finitely generated Free Modules, Torsion Elements of a Module, Torsion Submodule of a Module over a Commutative Domain, Noetherian Modules and Rings. Lecture 5, June 28. Rank of a Free Module, Structure Theorem of finitely generated modules over a PID, Hilbert Nullstellensatz. Lecture 6, June 29. Applications of the theory of finitely generated modules over PIDs to Linear Algebra, Discrete Valuations. Vikas Bist Panjab University, Chandigarh Lecture 1, July 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. Ring of endomorphisms on a vector space, Ring of formal power series. Lecture 2, July 2. Ring of Laurent power series, Twisted polynomial ring, Conditions for a twisted polynomial ring to be a domain, Simple algebras, Upper triangular matrix rings and their left ideals. Lecture 3, July 3. Endomorphism ring End D (V ) of an infinitely generated vector space over a division ring, Existence of a unique minimal ideal of End D (V ), Primitive rings, End D (V ) is primitive. Lecture 4, July 4. Wedderburn-Artin Theorem, Additive functions in End D (V ), Necessary conditions for End D 1 (V 1 ) to be isomorphic to End D 2 ( V 2), for finite dimensional vector spaces Vi over division rings D i . Lecture 5, July 5. Central algebras over a field K, Central simple K-algebras, Noether Skolem Theorem. Dinesh Khurana Panjab University, Chandigarh Lecture 1, July 1. Modules, Basic differences between vector spaces and modules, Existence of a free module with two bases with different cardinalities, Existence of rings in which left invertible elements are not right invertible. Lecture 2, July 2. 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. Lecture 3, July 3. Semisimple Modules and their various characterizations, Semisimple Rings and Wedderburn-Artin Structure Theorem. Lecture 4, July 4. Jacobson radical and its various characterizations, Characterization of semisimple rings as one-sided Artinian rings with zero Jacobson radical, R-balanced maps and their examples. Lecture 5, July 5. Tensor Products of Modules and Algebras. Lecture 6, July 6. Hom Functors and their exactness, Projective, Injective Modules, Characterization of Projective and Injective Z-modules. Lecture 7, July 6. Tensor Functors and their exactness, Flat Modules, Characterization of Flat Z-modules, von Neumann Regular Rings as generalization of semisimple rings and their characterization as rings over which every module is flat.

Time Table

 9.30 - 11.00Lecture 1 11.30-1.00Lecture 2 2.00- 4.00Tutorial June 24 MKD UC UC, MKD, PG June 25 UC MKD MKD, UC, PG June 26 UC MKD UC, MKD, PG June 27 UC MKD MKD, UC, PG June 28 UC MKD UC, MKD, PG June 29 UC MKD MKD, UC, PG July 1 VB DK VB, DK, AMB July 2 VB DK DK, VB, AMB July 3 VB DK VB, DK, AMB July 4 DK VB DK, VB, AMB July 5 DK VB VB, DK, AMB July 6 DK DK DK, VB, AMB

Tea/Snacks Breaks: 11.00 - 11.30 am and 4.00 - 4.30 pm
Lunch Break: 1.00 - 2.00 pm

List of actual Participants

How to reach

How to reach North-Eastern Hill University, Shillong.

From Guwahati Airport: One can take a shared tourist taxi from the airport to Police Bazar, Shillong, the cost is Rs. 500/- per person. From Police Bazar, one can directly hire a local cab upto NEHU Campus, the cost is about Rs. 200/-. Or one can go from Police Bazar to Polo Bazar and then take a shared taxi to the campus.

Alternately, one can get down at District Commissioner Office, near Wards Lake and take a university bus or a shared taxi from the point opposite to Raj Bhavan upto NEHU Campus.

From Guwahati Railway station: One can take a shared tourist taxi, the cost is Rs. 300/- per person or one can take a shared sumo or a bus upto Police Bazar and follow the same route as above.

From Shillong airport: There are local cabs available from Shillong airport upto Police Bazar, Shillong.

Contact Numbers for further assistance:

1. Dr. A.M. Buhphang – (O) 0364-2722720, (M) 9436161409, (M) 8414033829.

2. Mr. Soumitra Das – (M) 8974905463.