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
|
Learning Outcomes |
Learning and teaching strategies |
Assessment |
|---|---|---|
|
After the completion of the course the students will be able to: CLO102- 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. CLO103- Students get idea about types of modules. CLO104- 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. CLO105- Students know about Divisible groups, Jordon-Holder theorem, Indecomposable modules, Krull–Schmidt theorem, images and direct sum of semi-simple modules. CLO106- 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. |
Approach in teaching: Interactive Lectures, Discussion, Tutorials, Reading assignments, Demonstration, Team teaching Learning activities for the students: Self learning assignments, Effective questions, Simulation, Seminar presentation, Giving tasks, Field practical
|
Presentations by Individual Students Class Tests at Periodic Intervals. Written assignment(s) Semester End Examination |
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, 1989.
- T.Y.Lam, Noncommutative Rings, Springer-Verlag, 1991.
- I.N.Herstein, Noncommutative Rings, C. Monographs of AMS, 1968.
- T.W. Hungerford, Algebras, Springer, 1980.
- B. Hartley, T.O.Hauvkes, Rings, Modulesand Linear Algebra, Chapmann 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, 2003.
