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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.
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 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
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+Definition of a Noetherian and Artinian rings
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+Definitions of a local ring and quotient ring
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+Definition of the chain of prime ideals and the length of these chains
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+Definition of the Krull dimension (supremum of the lengh of chain of prime ideal) 
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+Definition of the height of prime ideal (dimension of A_p)
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+Give Examples of each of the above cases for a particular instances of ring
 
 Theorem 0: Hilbert Basis Theorem
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 Theorem 1: If A is a nonzero ring, then dim A[t] >= dim A +1
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 Theorem 2: If A is a nonzero noetherian ring, then dim A[t] = dim A + 1
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 Theorem 3: If A is nonzero ring then dim A_p + dim A/p <= dim A