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Definition 1 (monomial ideal)

[definition] An ideal $I \subseteq k[x_1, \ldots, x_n]$ is a monomial ideal if there is a subset $A \subseteq \ZZ_{\ge 0}^n$ such that $I$ consists of all polynomials which are of the form $\sum_{\alpha \in A} h_\alpha x^\alpha$, where $h_\alpha \in k[x_1, \ldots, x_n]$. In this case,

[/definition]

Lemma 2

[lemma] Let $I = \langle x^\alpha \mid \alpha \in A\rangle$ be a monomial ideal. Then a monomial $x^\beta$ lies in $I$ if and only if $x^\beta$ is divisible by $x^\alpha$ for some $\alpha \in A$. [/lemma]

Lemma 3

[lemma] Let $I$ be a monomial ideal, and let $f \in k[x_1, \ldots, x_n]$. TFAE:

1. $f \in I$.
2. Every term of $f$ lies in $I$.
3. $f$ is a $k$-linear combination of the monomials in $I$. [/lemma]

Corollary 4

[corollary] Two monomial ideals are the same if and only if they contain the same monomials. [/corollary]

Theorem 5 (Dickson’s Lemma)

[theorem] Let $I = \langle x^\alpha \mid \alpha \in A\rangle$ be a monomial ideal. Then $I$ admits a finite basis over $k$ using monomials in $A$. [/theorem]

Proposition 7

[proposition] A monomial ideal $I \subseteq k[x_1, \ldots, x_n]$ has a basis $x^{\alpha(1)}, \ldots, x^{\alpha(s)}$ with the property $x^{\alpha(i)}$ does not divide $x^{\alpha(j)}$ for $i \neq j$ (so an algebraically independent subset). Furthermore, this basis is unique and is called the minimal basis of $I$. [/proposition]