---
title: "Commutative Algebra: Main Results"
sort: 015
---

Tags: #study-guides 

# Results

- Why does $\tensor_R$ exist for $\mods{R}$, and how is it constructed?
- Primes and ideals:
	- What is [Krull's theorem](Krull's%20theorem.md)?
	- What is the [ideal correspondence for localizations](ideal%20correspondence%20for%20localizations.md)?
	- What is the [prime avoidance lemma](prime%20avoidance%20lemma.md)?
	- ✨What is [Nakayama's lemma](Unsorted/Nakayama's%20lemma.md)?
	- What is the [Artin-Rees theorem](Artin-Rees%20theorem.md)?
	- What is [Krull's intersection theorem](Unsorted/Krull's%20intersection%20theorem.md)?
	- What is [Krull's principal ideal theorem](Krull's%20principal%20ideal%20theorem.md)?
- What are the [Cohen–Seidenberg theorems](Cohen–Seidenberg%20theorems)?
	- What is the  [going up theorem](Unsorted/going%20up%20theorem.md)? 
	- What is the [going down theorem](going%20down%20theorem.md)?
	- What is the  [lying over theorem](lying%20over%20theorem.md)?
	-  What is the [incomparability theorem](incomparability%20theorem)?
- What is [Serre’s criterion](Serre’s%20criterion)?
- What is [Noether normalization](Noether%20normalization.md)?
- What is [Hensel's lemma](Hensel's%20lemma.md)?
- What is the [Cayley-Hamilon theorem for modules](Cayley-Hamilon%20theorem%20for%20modules.md)? 
- Toward [Algebraic Geometry Definitions](Projects/0000%20Flashcards/Algebraic%20Geometry%20Definitions.md):
	- What is [Zariski's lemma](Unsorted/Zariski's%20lemma.md)?
- What is the [Cohen structure theorem](Cohen%20structure%20theorem)?
- Hilbert's theorems:
	- What is the **weak Nullstensatz**? ("Theorem of zeros")
	- What is the (strong) **Nullstellensatz**?
	- What is the [Hilbert Basis Theorem](Unsorted/Hilbert%20Basis%20Theorem.md)?
	- What is the fundamental problem of [invariant theory](Unsorted/invariant%20theory.md)?
	- What is the [syzygy theorem](syzygy%20theorem)?
	- Why is the [Hilbert function](Hilbert%20function) a polynomial?
- [Fundamental SESs in commutative algebra](Fundamental%20SESs%20in%20commutative%20algebra.md)