Rings And Modules

ABOUT THE COURSE :This course is a self-contained elementary introduction to Rings and Modules. We will cover basic topics of Ring Theory and Module Theory which is a core course in Algebra. This course is compulsory to those who want to pursue research in Algebra. This is a 12 weeks course. In the first six weeks we focus on the Ring Theory part and the last six weeks we focus on the Module Theory part. PRE - REQUISITES :Group Theory, Linear AlgebraINTENDED-AUDIENCE:All learneres interested in Mathematics

WEEK 1 : Introduction to rings, examples, ideals, ring homomorphisms, quotients with examples WEEK 2 : Isomorphism theorems, Prime and maximal ideals, basic properties of ideals, local rings WEEK 3 : Radical ideals, Nil radical, Jacobson radical WEEK 4 : Integral domains, field of fractions, Euclidean Domains, Principal Ideal Domains WEEK 5 : Unique factorization domains, Gauss lemma, Eisenstein’s irreducibility criterion WEEK 6 : Algebraic variety, Weak form of Hilbert’s nullstellensatz WEEK 7 : Introduction to Modules over arbitrary rings, examples, Free modules, submodules, quotient modules, module homomorphisms, direct sums, direct products WEEK 8 : Generators of modules over commutative rings, rank of a free module, Simple modules WEEK 9 : Modules over a PID WEEK 10 : Modules with chain conditions, Noetherian modules, Noetherian rings WEEK 11 : Artinian modules, Artinian rings, Hilbert Basis Theorem WEEK 12 : Tensor product of modules, properties of tensor products