diff --git a/README.md b/README.md
index 9fe7965..8ffbe34 100644
--- a/README.md
+++ b/README.md
@@ -6,26 +6,49 @@ We start the commutative algebra project with a list of important definitions an
 
 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
+- Definitions of an ideal, prime ideal, and maximal ideal:
+```lean
+def Mathlib.RingTheory.Ideal.Basic.Ideal (R : Type u) [Semiring R] := Submodule R R
+class Mathlib.RingTheory.Ideal.Basic.IsPrime (I : Ideal α) : Prop
+class IsMaximal (I : Ideal α) : Prop
+```
 
-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
+class Mathlib.RingTheory.Artinian.IsArtinian (R M) [Semiring R] [AddCommMonoid M] [Module R M] : Prop
+```
+- Definition of a polynomial ring: `Mathlib.RingTheory.Polynomial.Basic`
 
-Definitions of a local ring and quotient ring
+- Definitions of a local ring and quotient ring: `Mathlib.RingTheory.Ideal.Quotient.?`
+```lean
+class Mathlib.RingTheory.Ideal.LocalRing.LocalRing (R : Type u) [Semiring R] extends Nontrivial R : Prop
+```
 
-Definition of the chain of prime ideals and the length of these chains
+- 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`
+
+- Krull dimension of a module
+
+- 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
+theorem Mathlib.RingTheory.Polynomial.Basic.Polynomial.isNoetherianRing [inst : IsNoetherianRing R] : IsNoetherianRing R[X]
+```
 
 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
+
+Lemma 0: A ring is artinian iff it is noetherian of dimension 0.
+
+Definition of a graded module