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@ -11,21 +11,28 @@ Feel free to add, modify, and expand this file. Below are starting points for th
def Mathlib.RingTheory.Ideal.Basic.Ideal (R : Type u) [Semiring R] := Submodule R R
```
- Definition of a Spec of a ring
- Definition of a Spec of a ring: `Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic.PrimeSpectrum`
- Definition of a Noetherian and Artinian rings
- Definition of a Noetherian and Artinian rings:
```lean
class Mathlib.RingTheory.Noetherian.IsNoetherian (R M) [Semiring R] [AddCommMonoid M] [Module R M] : Prop
```
- 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 Krull dimension (supremum of the lengh of chain of prime ideal): `Mathlib.Order.KrullDimension.krullDim`
- Definition of the height of prime ideal (dimension of A_p): `Mathlib.Order.KrullDimension.height`
- 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 0: Hilbert Basis Theorem:
```lean
instance isNoetherianRing [Finite σ] [IsNoetherianRing R] : IsNoetherianRing (MvPolynomial σ R)
```
Theorem 1: If A is a nonzero ring, then dim A[t] >= dim A +1