Tags: #qual_algebra

# Week 4: Rings

- Morphisms, Ideals, quotients, zero divisors, isomorphism theorems, CRT
- Irreducible and prime elements, nilpotent, units
- Radical, nilradical, spec and maxspec
- Special types: domains, integral domains, Euclidean ⇒ PID ⇒ UFD ⇒?, Dedekind domains, Noetherian, Artinian
- Zorn’s lemma arguments
- Bonus optional stuff: localization

__Seminars and Talks/Workshops/Algebra/_attachments/Untitled 15.png__

__Seminars and Talks/Workshops/Algebra/_attachments/Untitled 16.png__

__Seminars and Talks/Workshops/Algebra/_attachments/Untitled 17.png__

__Seminars and Talks/Workshops/Algebra/_attachments/Untitled 18.png__

Prove that a commutative ring with unit is a field if and only if its only ideals are {0} and the whole ring

__Seminars and Talks/Workshops/Algebra/_attachments/Untitled 19.png__

Show the irreducibility criterion for polynomials \(f\in k[x]\) of degree 2 or 3: such a polynomial is irreducible iff it has no roots in the field k

__Seminars and Talks/Workshops/Algebra/_attachments/Untitled 20.png__

__Seminars and Talks/Workshops/Algebra/_attachments/Untitled 21.png__