Module Theory

Module Theory (MAL7450)

January-April, 2024

Lectures : Tuesday at 3:00 PM, Wednesday and Friday at 2:00 PM.
Discussion/Office Hour : By Appointment

Evaluation :

Mid Semester Examinations - 30% (15%+15%)

End Semester Examination - 30%

Class Performance - 20%

Minor Project - 20%

Topics to be covered:

Modules over Commutative rings : Quick Review of Vector Spaces, Rings, Principal Ideal domains, and Examples, Definition and Examples of Modules, Quotient Modules and Module homomorphisms, Isomorphism Theorems, direct sums, finitely generated Modules, Free Modules and bases, Simple Modules, Semi simple Modules, Schur’s Lemma.Modules with chain conditions : Notherian and artinian modules and rings-Hilbert basis theorem, Wedderburn-Artin theorem, Wedderburn Decomposition of group algebras.Modules over a Principal Ideal Domain : Structure of finitely generated modules over PID : Cyclic Decomposition, Equivalence of matrices over PID, Finitely generated torsion modules, Classification of finitely generated modules over PID, Classification of Abelian groups.Modules over k[x] and linear operators : Review of linear operators on finite dimensional vector spaces, Canonical forms : Jordan Canonical form and Rational Canonical Form, and Applications.

References:

Musili, C., (1994), Introduction to rings and modules, 2nd Edition, Narosa.
Dummit ,D.S., Foote, R.M. (2004), Abstract algebra, 3rd Edition, Wiley.
Luther, I.S., Passi, I.B.S., (2013), Algebra; Volume 3: Modules, 1st Edition, Narosa.
Roman, S., (2008), Advanced linear algebra, 3rd Edition, Springer.
Lang, S., (2002), Algebra, Revised Third Edition, Springer.
Cohn, P. M. Algebra, Vols.I, II, III, John Wiley and Sons, 1982, 1989, 1991.

Course

3-Jan-2024 : Motivation and Introduction to the course
5-Jan-2024 : Definition of Module, Examples and Submodules

9-Jan-2024 : Left Module is same as additive abelian group M along with a ring homomorphism in to End(M)
10-Jan-2024 : No Class (Extra class will be scheduled to compensate)
12-Jan-2024 : Comparison of Free Modules and Vector Spaces

16-Jan-2024 : Quotient Modules and Free Modules over commutative ring with unity
17-Jan-2024 : Module Homomorphisms and Isomorphism Theorems
19-Jan-2024 : Simple Modules ( Class Test - 1 )

23-Jan-2024 : Basic results related to Modules over P.I.D
24-Jan-2024 : Modules with chain conditions
26-Jan-2024 : Republic Day