Commutative Algebra: Main Results

Why does \(\otimes_R\) exist for \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), and how is it constructed?
Primes and ideals:
- What is Krull's theorem?
- What is the ideal correspondence for localizations?
- What is the prime avoidance lemma?
- ✨What is Nakayama’s lemma?
- What is the Artin-Rees theorem?
- What is Krull’s intersection theorem?
- What is Krull's principal ideal theorem?
What are the Cohen--Seidenberg theorems?
- What is the going up theorem?
- What is the The Going Down Theorem?
- What is the lying over theorem?
- What is the incomparability theorem?
What is Serre's criterion?
What is Noether Normalization?
What is Hensel's lemma?
What is the Cayley-Hamilon theorem for modules?
Toward algebraic geometry:
- What is Zariski’s lemma?
- What is the weak Nullstensatz?
- What is the (strong) Nullstellensatz?
- What is the Hilbert Basis Theorem?
What is the Cohen structure theorem?