 Coefficient ideals of a polynomial ideal

For an ideal $K\subseteq R[t]$, let $A_n(K)$ consist of the coefficients at $t^n$ of elements of $K$ of polynomial degree at most $n$, including the zero polynomial. Each $A_n(K)$ is an ideal of $R$, and multiplication of representing polynomials by $t$ shows that $A_n(K)\subseteq A_{n+1}(K)$.

If $R$ is a Noetherian ring, this chain stabilizes at some $N$. Choose finite generating sets of an ideal for $A_0(K),\dots ,A_N(K)$, and for each nonzero generator choose a representing polynomial $f_{nj}\in K$ with that leading coefficient and polynomial degree $n$. These finitely many polynomials generate $K$: for any nonzero $f\in K$ of polynomial degree $d$, use the chosen generators at level $\min (d,N)$, multiplied by suitable powers of $t$, to cancel its leading coefficient. Induction on polynomial degree finishes the argument. This proves the Hilbert basis theorem without assuming that $R$ is an integral domain.