Modules and Rings-I
This course will enable the students to –
- Understand the importance of a ring as a fundamental object in algebra.
- It also gives ideas about module and represents fundamental algebraic structures used in abstract algebra.
- 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):
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Course |
Learning outcomes (at course level) |
Learning and teaching strategies |
Assessment Strategies |
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Course Code |
Course Title |
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MAT 325C |
Modules and Rings-I (Theory)
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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
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Quiz, Poster Presentations, Power Point Presentations, Individual and group projects, Open Book Test, Semester End Examination
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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.
Noetherian module and Artinian module, Composition series. Projective modules, Injective modules, Direct sum of projective modules, Direct product of injective modules.
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.
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.
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.
- 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.
- 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.
