I am doing a course in Commutative algebra and there is a lemma called Dickson's lemma which states the following:
Let $\mathfrak{I} = \langle X^{u}: u \in A\rangle$ for some set $A \subset \mathbb{N}^{n}$. Then $\exists$ $u_{1},u_{2},\dots u_{s}$ such that $\mathfrak{I} = \langle X^{u_{1}},X^{u_{2}},\dots, X^{u_{s}}\rangle$. Here $X^{u}$ is a monomial in the polynomial ring $k[x_{1}, \dots x_{n}]$ and $\mathfrak{I}$ is a monomial ideal.
So what Dickson's lemma is saying is that every monomial ideal in $k[x_1, \dots x_n]$ has a finite generating set. However I also looked at the definition given in Wikipedia and there Dickson's lemma is given as the following:
Dickson's lemma states that every set of $n$-tuples of natural numbers has finitely many minimal elements
I am sure they both mean the same thing, but I don't know why that is so and I can't link the definition given in Wikipedia to the one I have. Could someone explain in simple terms what Dickson's lemma really means and how it applies to the original definition I gave right at the start.
You should verify, that $\langle X^u | u \in A \rangle = \langle X^u | u \in B \rangle$ where $B \subset A$ is the set of minimal elements of $A$. Then the equivalence of the two formulations of Dickson's Lemma becomes clear, since the cited version states that $B$ is finite.