Modules and Rings-II

Paper Code: 
25MAT425(C)
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 
This course will enable the students to -
1. Explore a ring is an important fundamental concept in algebra and includes integers, polynomials and matrices as some of the basic examples. 
2. Analyze 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. Explore a module over a ring is a generalization of vector space over a field. 
Course Outcomes: 

Course

Learning outcomes

(vat course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

25MAT

425(C)

 

 

 

 

Modules and Rings-II

(Theory)

 

 

 

 

CO202: Explain local rings to prove basic properties of formal power series.

CO203: Determine semi simple modules and its characterization.

CO204: Explain simple ring, characterization of Artinian simple ring.

CO205: Analyze basic properties of the Jacobson radical, Jacobson Semisimple Rings, Hopkins-Levitzki Theorem, Nakayama's Lemma and regular ring.

CO206: Explore the concept of the lower and upper nil radical of a ring.

CO207: Contribute effectively in course-specific interaction.

 

Approach in teaching:

Interactive Lectures, Discussion, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions,  Topic  presentation, Assigned tasks

 

 

Quiz, Class Test, Individual projects,

Open Book Test, Continuous Assessment, Semester End Examination

 

 

 

Unit I: 
Local Ring:
15.00
 Local ring, Characterization of local ring, Local ring of formal power series.
 
Unit II: 
Semisimple Ring and Characterization:
15.00

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

Unit III: 
Simple Ring and Characterization: Simple ring, Characterization of Artinian simple ring.
15.00
 Simple ring, Characterization of Artinian simple ring.
 
Unit IV: 
Jacobson Ring:
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: 
Nil Radical:
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, Oxford University Press, London, 1990.
  • T.Y. Lam, Noncommutative Rings, Springer-Verlag, 2001. 
  • I.N. Herstein, Noncommutative Rings, C. The Mathematical Association of America, 2005.
  • T.W. Hungerford, Algebras, Springer, 2003.
  • B. Hartley, T.O. Hauvkes, Rings, Modules and Linear Algebra, Chapman and Hall Ltd., 1970.
 
References: 
  • R.B. Allenly, Rings Fields and Graphs: An Introduction of Abstract Algebra, Edward Arnold, 1991.
  • T.W. Hungerford, Algebras, Springer, 2003.
  • 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.
e- RESOURCES
 
JOURNALS
 
Academic Year: 
2025-2026
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