NOTE: Attempt any FIVE questions selecting at least TWO questions from each section.

Section-I (5/9)

Ring Theory

•    Construction of new rings

•    Direct sums, Polynomial rings

•    Matrix rings

•    Divisors, units and associates

•    Unique factorisation domains

•    Principal ideal domains and Euclidean domains

Field Extensions

•    Algebraic and transcendental elements

•    Degree of extension

•    Algebraic extensions

•    Reducible and irreducible polynomials

•    Roots of polynomials



Section-II (4/9)

Modules

•    Definition and examples

•    Submodules

•    Homomorphisms

•    Quotient modules

•    Direct sums of modules

•    Finitely generated modules

•    Torsion modules

•    Free modules

•    Basis, Rank and endomorphisms of free modules

•    Matrices over rings and their connection with the basis of a free module

•    A module as the direct sum of a free and a torsion module