MODULES AND RINGS-I (Optional Paper)
- 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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Paper Code |
Paper Title |
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MAT 325C
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Modules and Rings-I
(Theory)
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The students will be able to –
CO108: Students 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 the theory is presented with complete proofs. From the beginning the approach is categorical. CO109: Students get idea about types of modules. CO110: Students understand the structure theory of modules over a Euclidean domain along with its implications. 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. CO111: Students know about Divisible groups, Jordon-Holder theorem, Indecomposable modules, Krull–Schmidt theorem, images and direct sum of semi-simple modules. CO112: Students know about Prime ideals, m-system, Prime radical of an ideal, Prime radical of a ring, Semiprime ideal, n-system, Representation of a ring as a subdirect sum of rings, Subdirectly irreducible ring, Birkhoff theorem on subdirectly irreducible ring, Subdirectly irreducible boolean ring.
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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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Noetherian module and Artinian module, Composition series. Projective modules, Injective modules, Direct sum of projective modules, Direct product of injective modules.
- 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.
