Somehow Section 2

Definition 2.9 A monomial ideal J ⊂ k[x1, . . . , xn] is an ideal generated by a
collection of monomials {xα}α∈A.

Definition 2.10 Fix a monomial order > and let I ⊂ k[x1, . . . , xn] be an ideal.
The ideal of leading terms is defined
LT(I) := LT(g) : g ∈ I .
By convention, LT(0) = 0.

Definition 2.11 Fix a monomial order > and let I ⊂ k[x1, . . . , xn] be an ideal. A
Gr¨obner basis for I is a collection of nonzero polynomials
{ f1, . . . , fr} ⊂ I
such that LT( f1), . . . , LT( fr ) generate LT(I ).
Nothing in the definition says that a Gr¨obner basis actually generates I! We prove
this a posteriori.

Corollary 2.14 Fix a monomial order >. Let I ⊂ k[x1, . . . , xn] be an ideal and
f1, . . . , fr a Gr¨obner basis for I . Then I =  f1, . . . , fr .

Theorem 2.21 (Existence Theorem) Fix a monomial order > and an arbitrary
nonzero ideal I ⊂ k[x1, . . . , xn]. Then I admits a finite Gr¨obner basis for the prescribed
order.

Corollary 2.22 (Hilbert Basis Theorem) Every polynomial ideal is finitely
generated.
It suffices to showthat LT(I ) is finitely generated. Indeed, if f1, . . . , fr ∈ I are chosen
such that
LT(I ) = LT( f1), . . . , LT( fr )
then Corollary 2.14 implies
I =  f1, . . . , fr .
Thus the proof of the Existence Theorem is reduced to the case of monomial ideals:
Proposition 2.23 (Dickson’s Lemma) Every monomial ideal in a polynomial ring
over a field is generated by a finite collection of monomials.

Proposition 2.24 (Noether’s Proposition) Let R be a ring. Then the following
conditions are equivalent:
1. every ideal I ⊂ R is finitely generated;
2. every ascending chain of ideals
I0 ⊂ I1 ⊂ I2 ⊂ . . .
terminates, i.e., IN = IN+1 for sufficiently large N.
Then we say the ring R is Noetherian.

Theorem 2.26 Let R be a Noetherian ring. Then R[y] is also Noetherian.

Definition 2.27 The least common multiple of monomials xα and xβ is defined
LCM(xα, xβ ) = xmax(α1,β1)
1 . . . xmax(αn,βn )
n .