PhD-Algebra Students (first semester course)

This course will cover the following topics: 

Rings and Ideals: Rings and ring homomorphisms – Ideals and quotient rings – Zero divisors –

Nilpotent elements – units – Prime ideals and maximal ideals – Nilradical and Jacobson radical –

Operations on Ideals – Extension and contraction. Modules: modules and module

homomorphisms – submodules and quotient modules – operations on submodules – direct sum

and direct product – finitely generated modules – exact sequences – tensor product of modules –

restriction and extension of scalars – exactness properties of the tensor product – algebras –

tensor product of algebras. Rings and modules of fractions: Local properties – extended and

contracted ideals in rings of fractions. Primary decomposition. Integral dependence and

valuations: The Going up theorem – integrally closed integral domains - The Going down

theorem – Valuation of rings. Chain conditions. Noetherian rings: Primary decomposition in

Noetherian rings. Artin rings, Dimension theory.