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 - 11.00
Lecture 1
11.30-1.00
Lecture 2
2.00- 4.00
Tutorial
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

Sr. SID Full Name Gender Affiliation Univ/Inst.
M.Sc./M.A.
Year of Passing M.Sc./M.A Ph.D. Deg. Date
1 23904 Mr. Surya Prakash Male CMP Degree College University of Allahabad University of Allahabad 2015  
2 23905 Mr. Ajim Uddin Ansari Male CMP Degree College University of Allahabad University of Allahabad 2015  
3 23908 Dr. Mohammad Salahuddin Khan Male Aligarh Muslim University Aligarh Muslim University 2007 03/12/2013
4 23953 Dr. Abdul Nadim Khan Male King Abdulaziz University MJP Rohilkhand University 2009 01/11/2014
5 23966 Dr. Avanish Kumar Chaturvedi Male University of Allahabad V B S P U Jaunpur 2003 25/07/2009
6 23984 Dr. Uday Shankar Chakraborty Male Assam University, Silchar Assam University, Silchar 2004 27/01/2010
7 24097 Dr. Dheerendra Mishra Male The LNM Institute of Information Technology, Jaipur Jiwaji University, Gwalior   03/04/2014
8 24226 Ms. Jane Roseline Yimchunger Female Kohima Science College North-Eastern Hill University 1999  
9 24227 Dr Aftab Hussain Shah Male Central University of Kashmir AMU Aligarh 2004 12/04/2010
10 24236 Dr. Aditya Mani Mishra Male Rajasthan Technical University Kota Motilal Nehru NIT Allahabad 2010 18/05/2014
11 24426 Mr. Sanjeev Kumar Maurya Male Adjunct Faculty, College of Engineering Pune VBS Purvanchal University, Jaunpur 2010  
12 24435 Dr. Krishnendu Das Male Netaji Subhas Mahavidyalaya University of Hyderabad 2009 14/08/2018
13 24655 Dr Sajad Hussain Ahangar Male Government Degree College Shopian University of Kashmir Srinagar 2007 27/06/2016
14 25716 Mr Khwairakpam Herachandra Singh Male North Eastern Regional Institute of Science and Technology NEHU, Shillong 2003 15/03/2014
15 26117 Dr. Manoj Kumar Patel Male National Institute of Technology Nagaland VBS Poorvanchal University Jaunpur 2004 01/07/2011
16 26522 Dr. Paltu Sarkar Male University of North Bengal IIT KHARAGPUR 2008 05/12/2017
17 26911 Mr. Rahul Arora Male Malwa College S.C.D Government College 2008  
18 26978 Asst. Prof Soumen Pradhan Male Nagar College I.I.T. Kanpur 2012  
19 27053 Mr. Ranjan Sarkar Male Hijli college University of kalyani 2006  
20 27201 Mr. Sedevikho Chase Male Kohima Science College Dibrugarh University 2008  
21 24411 Mr Laithun Boro Male NEHU MS 2017  
22 27342 Ms . Mridusmita Hazarika Female North Eastern Hill University north eastern hill university    
23 25814 Mr. Soumitra Das Male North-Eastern Hill University North-Eastern Hill University 2012  
24 26350 Mr. Ngangom Rojitkumar Singh Male NEHU MANIPUR UNIVERSITY 2012  
25 26244 Mr Kamakhya Paul Male NEHU PONDICHERRY UNIVERSITY 2015  
26 26251 Mr. Pravesh Sharma Male NEHU,Shillong NEHU 2016  
27 26559 Mr. Champak Talukdar Male NEHU Gauhati University 2017  
28 26728 Ms. Dimpy Mala Dutta Female North Eastern Hills University Tezpur University 2017  
29 26891 Mr. Snehashish Sarkar Male North eastern hill university Cotton university 2017  
30 26910 Mr Shanborlang Bynnud Male North Eastern Hills University North Eastern Hills University, Shillong 2016  
31 27309 Ms Vitsono Lungalang Female Kohima Science College North Eastern Hill University 2003  
32 24672 Mr. Sushobhan Maity Male Visva-Bharati Visva-Bharati 2013  
33 24056 Mr. Prajnan Kumar Bhagawati Male Gauhati University Pondicherry University 2017  
34 24180 Ms Padmapriya V P Female Central University of Kerala MG University, Kottayam 2014  
35 26440 Ms. Raneeta Dutta Female IISER Bhopal IIT Chennai 2016  
36 24964 Mr. Shiv Kumar Male IIT(B.H.U.) Varanasi M.J.P. Rohilkhand University Campus Bareilly (U.P.) 2014  
37 26717 Mr. Sanjay Singha Male Indian Institute of Information Technology Vadodara VISVA-BHARATI  2018

 


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.