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SLMath collaboration for adding Krull dimension and Hilbert polynomial to mathlib
We start the comm algebra project by important definitions and theorems and go from there.
Feel free to add, modify, and expand this file. Below are starting point 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
Definition of a local ring and quotient ring
Definition of the Krull dimension
Give examples of each of the above cases for a particular instances of ring
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