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Modules and Rings-I [1]

Paper Code: 
24MAT325(C)
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to –

  1. Test the importance of a ring as a fundamental object in algebra.
  2. Integrate the ideas about module and represents fundamental algebraic structures used in abstract algebra. 
  3. Create a module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring (with identity) and a multiplication (on the left and/or on the right) is defined between elements of the ring and elements of the module.

 

Course Outcomes: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

24MAT

325(C)

 

 

 

 

 Modules and Rings-I

   (Theory)

 

 

 

 

CO131: Analyze the types of modules and applications of their lemmas.

CO132: Formulate the concepts of Noetherian modules, Artinian modules, projective modules and injective module.

CO133: Explore properties and examples of divisible groups, techniques for constructing embedding’s of modules.

CO134: Analyze Prime ideals, m-system, Prime radical of an ideal, Prime radical of a ring, Semi-prime ideal, n-system.

CO135: Explore applications of sub direct sums of rings in ring theory; understand the statement and proof of Birkhoff's theorem.

CO136: 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: 
Modules I:
15.00

Morphisms, Exact sequences, The three lemma, The four lemma, The five lemma, Butterfly of zausenhauss theorem, Product and co-product of R-modules, Free modules.

 

Unit II: 
Modules II:
15.00

Noetherian module and Artinian module, Composition series, Projective modules, Injective modules, Direct sum of projective modules, Direct product of injective modules.

 

Unit III: 
Modules III:
15.00

Divisible groups, Embedding of a module in an injective module, Tensor product of modules, Noetherian module and Artinian module, Finitely generated modules, Jordon-Holder theorem, Indecomposable modules, Krull–Schmidt theorem, Semi-simple modules, Submodules, Homomorphic images and direct sum of semi-simple modules.

 

Unit IV: 
Ideals and Rings:
15.00

Prime ideals, m-system, Prime radical of an ideal, Prime radical of a ring, Semiprime ideal, n-system, Prime rings, Semiprime ring as a subdirect product of a prime ring, Prime ideals and prime radical of matrix ring.

 

Unit V: 
Subdirect Rings:
15.00

Subdirect sum of rings, Representation of a ring as a subdirect sum of rings, Subdirectly irreducible ring, Birkhoff theorem on subdirectly irreducible ring, Subdirectly irreducible Boolean ring.

 

Essential Readings: 
  • T. S. Blyth, Module Theory, Clarendon Press, London, 1990.
  • T. Y. Lam, Non commutative Rings, Springer-Verlag, 2001.
  • I. N. Herstein, Non commutative Rings, The Mathematical Association of America, 2005.
  • T. W. Hungerford, Algebra, Springer, 2003.

SUGGESTED READING

  •    B. Hartley and T.O. Hauvkes, Rings, Modules and Linear Algebra, Chapman and Hall Ltd., 1970.
  • R. B. Allenly, Rings Fields and Graphs: An Introduction of Abstract Algebra, Edward Arnold, 1989.
  • T. W. Hungerford, Algebras, Springer, 2003.
  • J. Rose, A Course on Ring Theory, Cambridge University Press, 2003.
  • 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.

e- RESOURCES

 

JOURNALS

 

Academic Year: 
2024-2025 [9]

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Source URL: https://maths.iisuniv.ac.in/courses/subjects/modules-and-rings-i-2

Links:
[1] https://maths.iisuniv.ac.in/courses/subjects/modules-and-rings-i-2
[2] https://nptel.ac.in/courses/111106131
[3] https://nptel.ac.in/courses/111106098
[4] https://nptel.ac.in/courses/111102009
[5] http://www.uop.edu.pk/ocontents/Ring%20theory.pdf
[6] https://www.sciencedirect.com/journal/journal-of-pure-and-applied-algebra/vol/133/issue/1
[7] https://www.mdpi.com/journal/symmetry/special_issues/Commutative_Ring_Theory
[8] https://www.academia.edu/Documents/in/Ring_Theory
[9] https://maths.iisuniv.ac.in/academic-year/2024-2025