Here's why I think this:
Let $R=\langle a_1,\dots,a_m\rangle$ (I'm assuming $R$ is finitely generated because that's all I'm concerned with), then each element of $R[x]$ can be written as $$(r_{1,0}a_1+r_{2,0}a_2+\cdots+r_{m,0}a_m)+(r_{1,1}a_1+r_{2,1}a_2+\cdots+r_{m,1}a_m)x+\cdots+(r_{1,n}a_1+r_{2,n}a_2+\cdots+r_{m,n}a_m)x^n$$ $$=(r_{1,0}+r_{1,1}x+\cdots+r_{1,n}x^n)a_1+(r_{2,0}+r_{2,1}x+\cdots+r_{2,n}x^n)a_2+\cdots+(r_{m,0}+r_{m,1}x+\cdots+r_{m,n}x^n)a_m,$$ which is a sum of elements of $\{a_1,\dots,a_m\}$ with coefficients in $R[x]$, so $R[x]=\langle a_1,\dots,a_m\rangle$.
Here's why I think this is wrong:
It trivially implies the Hilbert basis theorem, which can't be that easy to prove.