# Commutative algebra in Lean

Welcome to the repository for adding definitions and theorems related to Krull dimension and Hilbert polynomials to mathlib.

We start the commutative algebra project with a list of important definitions and theorems and go from there.

Feel free to add, modify, and expand this file. Below are starting points for the project:

Definitions of an ideal, prime ideal, and maximal ideal

Definition of a Spec of a ring

Definition of a Noetherian and Artinian rings

Definitions of a local ring and quotient ring

Definition of the chain of prime ideals and the length of these chains

Definition of the Krull dimension (supremum of the lengh of chain of prime ideal) 

Definition of the height of prime ideal (dimension of A_p)

Give Examples of each of the above cases for a particular instances of ring

Theorem 0: Hilbert Basis Theorem

Theorem 1: If A is a nonzero ring, then dim A[t] >= dim A +1

Theorem 2: If A is a nonzero noetherian ring, then dim A[t] = dim A + 1

Theorem 3: If A is nonzero ring then dim A_p + dim A/p <= dim A