Syllabus
1. Basic notions (maximal ideals, prime ideals, prime radical, fractional ideals, divisors).
2. Integral extensions (closures, quotient rings and polynomials, extension of homomorphisms).
3. Valoation domains (basic properties, integral closure, basic constructions, power series, domains finitely generated over fields).
4. Noetherian rings (basic properties, Artin-Rees Theorem, primary decomposition).
5. Dedekind domains (invertible ideals, Dedekind domains, Dedekind rings).
6. Integral closures of noetherian domains (separable case, Krull-Akudzuki Theorem).
Annotation
Integral extensions, valuation domains, noetherian rings (Artin-Rees theorem), Dedekind domains, integral closures of noetherian domains (separable case, Krull-Akizuki theorem).
The knowledge of the material of the course Algebra II (NALG027) is desirable.