MODULES AND RINGS-II (Optional Paper)

Paper Code: 
MAT425C
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 
This course will enable the students to -
  1. Understanding a ring is an important fundamental concept in algebra and includes integers, polynomials and matrices as some of the basic examples. 
  2. Understand the study of modules over a ring R provides students with an insight into the structure of R. 
  3. Develop ring and module theory leading to the fundamental theorems of Wedderburn and some of its applications.
  4. Understanding a module over a ring is a generalization of vector space over a field. 

Learning Outcomes

Learning and teaching strategies

Assessment

After the completion of the course the students will be able to:

CLO151- Understand the concept of a module as a generalisation of a

vector space and an Abelian group.

CLO152- Constructions such as direct sum, product and tensor product,Simple modules, Semisimple modules, artinian modules, their endomorphisms and examples.

CLO153- Radical, simple and semisimple artinian rings, examples and the Artin-Wedderburn theorem.

CLO154- The concept of central simple algebras, the theorems of Wedderburn and Frobenius.

CLO155- Student will understand The Jacobson radical, Jacobson

radical of matrix ring, Jacobson semisimple ring.

Approach in teaching:

Interactive Lectures, Discussion, Tutorials, Reading assignments, Demonstration, Team teaching

 

Learning activities for the students:

Self learning assignments, Effective questions, Simulation, Seminar presentation, Giving tasks, Field practical

 

Presentations by Individual Students

Class Tests at Periodic Intervals.

Written assignment(s)

Semester End Examination

 

Unit I: 
I
15.00

Local ring, Characterization of local ring, Local ring of formal power series.

Unit II: 
II
15.00

Semisimple module, Semisimple ring, Characterizations of semisimple module and semisimple ring, Wedderburn-Artin theorem on semisimple ring.

Unit III: 
III
15.00

Simple ring, Characterization of Artinian simple ring.

Unit IV: 
IV
15.00

The Jacobson radical, Jacobson radical of matrix ring, Jacobson semisimple ring, Relation between Jacobson semisimple ring and semisimple ring, Hopkins-Levitzki theorem, Nakayama’s lemma, Regular ring, Relation among semisimple ring, Regular ring and Jacobson semisimple ring.

 

Unit V: 
V
15.00

Lower nil radical, Upper nil radical, Nil radical, Brauer’s lemma, Kothe’s conjecture, Levitzki theorem.

Essential Readings: 
  • T.S.Blyth,Module Theory, Clarendon Press, London, 1989.
  • T.Y.Lam,Noncommutative Rings, Springer-Verlag, 1991.
  • I.N.Herstein,Noncommutative Rings, C. Monographs of AMS, 1968.
  • T.W. Hungerford,Algebras, Springer, 1980.
References: 
  • B. Hartley, T.O.Hauvkes, Rings,Modulesand Linear Algebra, Chapmann and Hall Ltd., 1970.
  • R.B.Allenly, Rings Fields and Graphs: An Introduction of Abstract Algebra,Edward Arnold,1989.
  • T.W. Hungerford,Algebras, Springer, 1980.
  • J.Rose, A Course on Ring Theory, Cambridge University Press, 1978.
  • L.H.Rowen, Ring Theory (Student Addition),Academic Press, 1991.
  • N.Jacobson, Structure of Rings, AMS, 1970,
  • P.M. Cohn, Basic Algebra,Springer; Corrected 2003. edition 2002.

 

Academic Year: 
2019-2020
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