Hilbert's basis theorem

Suppose ${\displaystyle {\mathfrak {a}}\subseteq R[X]}$ is a non-finitely generated left ideal. Then by recursion (using the axiom of dependent choice) there is a sequence of polynomials ${\displaystyle \{f_{0},f_{1},\ldots \}}$ such that if ${\displaystyle {\mathfrak {b}}_{n}}$ is the left ideal generated by ${\displaystyle f_{0},\ldots ,f_{n-1}}$ then ${\displaystyle f_{n}\in {\mathfrak {a}}\setminus {\mathfrak {b}}_{n}}$ is of minimal degree. By construction, ${\displaystyle \{\deg(f_{0}),\deg(f_{1}),\ldots \}}$ is a non-decreasing sequence of natural numbers. Let ${\displaystyle a_{n}}$ be the leading coefficient of ${\displaystyle f_{n}}$ and let ${\displaystyle {\mathfrak {b}}}$ be the left ideal in ${\displaystyle R}$ generated by ${\displaystyle a_{0},a_{1},\ldots }$. Since ${\displaystyle R}$ is Noetherian the chain of ideals

${\displaystyle (a_{0})\subset (a_{0},a_{1})\subset (a_{0},a_{1},a_{2})\subset \cdots }$

must terminate. Thus ${\displaystyle {\mathfrak {b}}=(a_{0},\ldots ,a_{N-1})}$ for some integer ${\displaystyle N}$. So in particular,

${\displaystyle a_{N}=\sum _{i<N}u_{i}a_{i},\qquad u_{i}\in R.}$

Now consider

${\displaystyle g=\sum _{i<N}u_{i}X^{\deg(f_{N})-\deg(f_{i})}f_{i},}$

whose leading term is equal to that of ${\displaystyle f_{N}}$; moreover, ${\displaystyle g\in {\mathfrak {b}}_{N}}$. However, ${\displaystyle f_{N}\notin {\mathfrak {b}}_{N}}$, which means that ${\displaystyle f_{N}-g\in {\mathfrak {a}}\setminus {\mathfrak {b}}_{N}}$ has degree less than ${\displaystyle f_{N}}$, contradicting the minimality.

Let ${\displaystyle {\mathfrak {a}}\subseteq R[X]}$ be a left ideal. Let ${\displaystyle {\mathfrak {b}}}$ be the set of leading coefficients of members of ${\displaystyle {\mathfrak {a}}}$. This is obviously a left ideal over ${\displaystyle R}$, and so is finitely generated by the leading coefficients of finitely many members of ${\displaystyle {\mathfrak {a}}}$; say ${\displaystyle f_{0},\ldots ,f_{N-1}}$. Let ${\displaystyle d}$ be the maximum of the set ${\displaystyle \{\deg(f_{0}),\ldots ,\deg(f_{N-1})\}}$, and let ${\displaystyle {\mathfrak {b}}_{k}}$ be the set of leading coefficients of members of ${\displaystyle {\mathfrak {a}}}$, whose degree is ${\displaystyle \leq k}$. As before, the ${\displaystyle {\mathfrak {b}}_{k}}$ are left ideals over ${\displaystyle R}$, and so are finitely generated by the leading coefficients of finitely many members of ${\displaystyle {\mathfrak {a}}}$, say

${\displaystyle f_{0}^{(k)},\ldots ,f_{N^{(k)}-1}^{(k)}}$

with degrees ${\displaystyle \leq k}$. Now let ${\displaystyle {\mathfrak {a}}^{*}\subseteq R[X]}$ be the left ideal generated by:

${\displaystyle \left\{f_{i},f_{j}^{(k)}\,:\ i<N,\,j<N^{(k)},\,k<d\right\}\!\!\;.}$

We have ${\displaystyle {\mathfrak {a}}^{*}\subseteq {\mathfrak {a}}}$ and claim also ${\displaystyle {\mathfrak {a}}\subseteq {\mathfrak {a}}^{*}}$. Suppose for the sake of contradiction this is not so. Then let ${\displaystyle h\in {\mathfrak {a}}\setminus {\mathfrak {a}}^{*}}$ be of minimal degree, and denote its leading coefficient by ${\displaystyle a}$.

Case 1: ${\displaystyle \deg(h)\geq d}$. Regardless of this condition, we have ${\displaystyle a\in {\mathfrak {b}}}$, so ${\displaystyle a}$ is a left linear combination

${\displaystyle a=\sum _{j}u_{j}a_{j}}$

of the coefficients of the ${\displaystyle f_{j}}$. Consider

${\displaystyle h_{0}=\sum _{j}u_{j}X^{\deg(h)-\deg(f_{j})}f_{j},}$

which has the same leading term as ${\displaystyle h}$; moreover ${\displaystyle h_{0}\in {\mathfrak {a}}^{*}}$ while ${\displaystyle h\notin {\mathfrak {a}}^{*}}$. Therefore ${\displaystyle h-h_{0}\in {\mathfrak {a}}\setminus {\mathfrak {a}}^{*}}$ and ${\displaystyle \deg(h-h_{0})<\deg(h)}$, which contradicts minimality.

Case 2: ${\displaystyle \deg(h)=k<d}$. Then ${\displaystyle a\in {\mathfrak {b}}_{k}}$ so ${\displaystyle a}$ is a left linear combination

${\displaystyle a=\sum _{j}u_{j}a_{j}^{(k)}}$

of the leading coefficients of the ${\displaystyle f_{j}^{(k)}}$. Considering

${\displaystyle h_{0}=\sum _{j}u_{j}X^{\deg(h)-\deg(f_{j}^{(k)})}f_{j}^{(k)},}$

we yield a similar contradiction as in Case 1.

Thus our claim holds, and ${\displaystyle {\mathfrak {a}}={\mathfrak {a}}^{*}}$ which is finitely generated.

Note that the only reason we had to split into two cases was to ensure that the powers of ${\displaystyle X}$ multiplying the factors were non-negative in the constructions.