Modules and Rings-II

Paper Code: 
MAT 425C
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. 

 

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Paper Code

Paper Title

 

 

MAT 425C

 

 

 

 

 

 

 

 

Modules and Rings-II

(Theory)

 

 

 

 

The students will be able to –

 

CO200: Understand the concept of a module as a generalisation of a vector space and an Abelian group.

CO201: Constructions such as direct sum, product and tensor product, Simple modules, Semisimple modules, artinian modules, their endomorphisms and examples.

CO202: Radical, simple and semisimple artinian rings, examples and the Artin-Wedderburn theorem.

CO203: The concept of central simple algebras, the theorems of Wedderburn and Frobenius.

CO204: Student will understand The Jacobson radical, Jacobson radical of matrix ring, Jacobson semisimple ring.

CO205: Apply the concepts of Simple modules, Semisimple modules, artinian modules, their endomorphisms for solving the real life problems.

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Field trips

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, 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.
  • B. Hartley, T.O.Hauvkes, Rings, Modules and 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: 
2021-2022
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