Noetherian is polynomial-closed

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 $$0 \le j \le N$$, 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:

$$\bigcup_{0 \le j \le N} S_j$$

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.