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.
- Teacher: Mulugeta Habte