Noetherian is polynomial-closed

This article gives the statement, and possibly proof, of a commutative unital ring property satisfying a commutative unital ring metaproperty
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Statement

Property-theoretic statement

The property of being a Noetherian ring is a polynomial-closed commutative unital ring property.

Verbal statement

If a ring is Noetherian, so is the polynomial ring in one variable over it.

Proof

The proof of this statement is as follows. Let $R$ be a Noetherian ring, and $I$ an ideal of $R[x]$. We need to prove that $I$ is finitely generated.

Consider $L_{n}(I)$ to be the set of leading coefficients of elements of $I$ which are polynomials of degree at most $n$. (the leading coefficient of a polynomial is the coefficient of the highest degree term with nonzero coefficient, and the leading coefficient of the zero polynomial is taken as zero). It is clear that each $L_{n}(I)$ is an ideal in $R$.

Further, since $R$ is Noetherian, the ascending chain:

$L_{0}(I)\subseteq L_{1}(I)\subseteq \ldots$

must stabilize at some stage, say $L_{N}(I)$.

Now, for each $L_{j}(I)$ with $0\leq j\leq N$, pick a finite generating set and take a representative polynomial for each generator. Call the set of representatives for degree $j$ as $S_{j}$. Then, the claim is that:

$\bigcup _{0\leq j\leq N}S_{j}$

generates the whole of $I$.

Suppose not. Then there exists a polynomial of minimal degree $m$ not generated by the union of the above. Now, we can construct a polynomial in the ideal generated by $S_{j}$, of the same degree $m$, and with the same leading coefficient: This is done by the fact that its leading coefficient lies in the ideal generated by leading coefficients of elements of $S_{j}$s. Taking the difference of these polynomials, gives a polynomial of smaller degree, which lies in the ideal if and only if the original polynomial lies in the ideal. This contradicts minimality of $n$.

Observations

Notice that a similar proof cannot be used to conclude that the polynomial ring over a principal ideal ring must be a principal ideal ring.

The problem here is that even though the ideal of leading coefficients may be generated by a single element, that single element may occur at a high degree, and the filtration $L_{n}(I)$ may include lots of other polynomials for smaller degree generators.