Modules and Rings-I

Paper Code: 
MAT325C
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 
This course will enable the students to –
  1. Understand the importance of a ring as a fundamental object in algebra.
  2. It also gives ideas about module and represents fundamental algebraic structures used in abstract algebra. 
  3. Understand 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 (COs):

 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Paper Code

Paper Title

MAT 325C

 

 

 

Modules and Rings-I

   (Theory)

 

 

 

 

The students will be able to –

 

CO128: Get an introduction to module theory and the related part of ring theory starting from a basic understanding of linear Class Tests at Periodic algebra.

CO129: Describe the types of modules.

CO130: Understand the structure theory of modules over a Euclidean domain along with its implications.

CO131: Know about Divisible groups, Jordon-Holder theorem, Indecomposable modules, Krull–Schmidt theorem, images and direct sum of semi-simple modules.

CO132: Identify Prime ideals, m-system, Prime radical of an ideal, Prime radical of a ring, Semiprime ideal, n-system.

CO133: The material underpins many later courses in algebra and number theory, and thus should give students a good background for studying these more advanced topics.

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
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: 
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: 
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: 
IV
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: 
V
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, 1989.
  • T. Y. Lam, Non commutative Rings, Springer-Verlag, 1991. 
  • I. N. Herstein, Non commutative Rings, C. Monographs of AMS, 1968.
  • T. W. Hungerford, Algebras, Springer, 1980.
  • 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, 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.
 
Academic Year: 
2021-2022
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